Download Mathematical Foundations for Computer Science I B.sc., IT

Mathematical Foundations for Computer Science
I B.sc., IT
DISCRETE MATHEMATICS ( I B.SC COMPUTER SCIENCE )
Unit-I
Matrix, Determinants and Inverse
Contents:
1.0 Aims and objectives
1.1 Introduction
1.2 Matrices
1.2.1 Applications of matrices
1.2.2 Definition of a Matrix
1.2.3 Matrix types
1.3 Determinant
1.3.1 Definition
1.3.2 Properties of determinants
1.3.3 Determinant of a 3 x 3 matrix by Matrix Enhancement
1.3.4 Leibnitz formula to compute determinant of a Matrix
1.3.5 Laplace's formula to compute determinant of a Matrix
1.4 Inverse of a Matrix
1.4.1 Cofactors
1.4.2 Adjoint of a Matrix
1.4.3 Inverse of a Matrix – Definition
1.4.4 Properties of invertible matrices
1.4.5 Matrix inverses in real-time simulations
1.4.6 Analytic solution
1.5 Problems and Solutions
1.6 Let Us Sum Up
1.7 Lesson End Activities
1.8 References
1.0 Aims and objectives
In this lesson, we have discussed the representation of data using matrices and the
representation of expressions using determinant. Matrix inversion, a convenient way to
solve system of linear equations, is also discussed in this lesson.
After reading this lesson, you should able to understand
Matrix and its usage in data representation.
Determinant and the various methods of computing it.
Cofactors and Adjoint of a matrix.
How to find matrix Inverse? And to solve linear equations using inverse.
Properties of determinant and invertible matrices.
Applications of Matrix inverses in real-time simulations
1.1 Introduction
J. J. Sylvester coined the term “matrix” in 1848. A matrix is a rectangular table of
elements. Matrices are used to describe linear equations, keep track of the coefficients of
linear transformation and to record data that depend on multiple parameters. The
determinant notation is now employed in almost every branch of applied science.
Matrix inversion plays a significant role in computer graphics and to solve system of
linear equations.
1.2 Matrices
The study of matrices is quite old. A 3-by-3 magic square appears in Chinese
literature dating from as early as 650 BC. Matrices have a long history of application in
solving linear equations. After the development of the theory of determinants by Seki
Kowa and Leibniz in the late 17th century, Cramer developed the theory further in the
18th century, presenting Cramer's rule in 1750. Carl Friedrich Gauss and Wilhelm Jordan
developed Gauss-Jordan elimination in the 1800s. Cayley, Hamilton, Grassmann,
Frobenius and von Neumann are among the famous mathematicians who have worked on
matrix theory. Olga Taussky-Todd (1906-1995) used matrix theory to investigate an
aerodynamic phenomenon called fluttering or aeroelasticity during WWII.
In mathematics, a matrix (plural matrices) is a rectangular table of elements (or
entries), which may be numbers or, more generally, any abstract quantities that can be
added and multiplied. Matrices are used to describe linear equations, keep track of the
coefficients of linear transformation and to record data that depend on multiple
parameters. Matrices can be added, multiplied, and decomposed in various ways, making
them a key concept in linear algebra and matrix theory.
1.2.1 Applications of matrices
Encryption
Matrices can be used to encrypt numerical data. Multiplying the data
matrix with a key matrix does encryption. Simply multiplying the encrypted
matrix with the inverse of the key does decryption.
Computer graphics
4×4 transformation matrices are commonly used in computer graphics.
The upper left 3×3 portion of a transformation matrix is composed of the new X,
Y, and Z-axes of the post-transformation coordinate space.
1.2.2 Definition of a Matrix
If m and n are positive integers, then an m n matrix (read “m b y n”) is a
rectangular array
in which each entry, ai,j, of the matrix is a real number. An m n matrix has m rows
(horizontal lines) and n columns (vertical lines).
m rows
aaaa
aaaa
aaaa
aaaa
n columns
A matrix having m rows and n columns is said to be of order m n. If m = n, the
matrix is square of order n. For a square matrix, the entries a11, a22, a33,…, ann are the
main diagonals.
A matrix that has only one row is a row matrix, and a matrix that has only one
column is a column matrix.
1.2.3 Matrix types
1) Diagonal Matrix
A square matrix A is said to be a diagonal matrix if aij= 0 when i j. In a
diagonal matrix all the entries except the entries along the main diagonal is
zero.
2) Triangular matrix
A square matrix in which all the entries above the main diagonal are zero
is called a lower triangular matrix. If all the entries below the main diagonal
are zero, it is called an upper triangular matrix.
3) Scalar Matrix
A scalar matrix is a diagonal matrix in which all the entries along the main
diagonal are equal.
4) Identity Matrix or Unit Matrix
An Identity Matrix or Unit Matrix is a scalar matrix in which entries along
the main diagonal are equal to 1.
5) Zero or Null or Void Matrix
In which all the entries are zero.
6) Equality of matrices
The matrices A=[aij]mxn and B=[bij]pxq are equal if m = p, n = q and aij = bij
for every i and j.
1.3 Determinant
Many complicated expressions of electrical and mechanical systems can be
conveniently handled by expressing them in “determinant form”.
1.3.1 Definition
In algebra, a determinant is a function depending on n that associates a scalar, det(A) or
, to every n × n square matrix A. The fundamental geometric meaning of a determinant
is as the scale factor for volume when A is regarded as a linear transformation.
Determinants are important both in calculus, where they enter the substitution rule for
several variables, and in multilinear algebra.
(or)
The mathematical expression a1b2-a2b1 can be expressed as
The above form is called a determinant of the second order. Each of the numbers a1, b1, a2,
b2 is called an element of the determinant.
Thus
Similarly, the arrangement of nine elements as
is called a determinant of the third order.
Thus
In general, a determinant of the nth order is denoted by
The determinant in (1) have n rows and n columns; thus having n2 elements. The diagonal
through the left hand top corner which contains the elements a1, b2, c3, ---, ln is called the
leading or principal diagonal.
1.3.2 Properties of determinants
Changing the rows into columns or columns into rows does not affect the value
of a determinant.
Interchanging a pair of rows (or columns) of matrix A, changes the sign of det(A).
If matrix A has a pair of identical rows (or columns) then det(A)=0
If each element in a row (or column) of matrix A is multiplied by k, then det(A) is
multiplied by k.
Note: Remember that when a matrix is multiplied by a scalar k, every element
must be multiplied by k.
If the elements of one row (or column) of matrix A are multiples of the elements
of another row(or column), then det(A)=0.
If each element of one row (or column) of matrix A is replaced by a new element
consisting of the original element plus a multiple of the corresponding element
from another row (or column), then the value of det(A) is unchanged.
det(A)=det(AT)
det(AB)=det(A) det(B)
1.3.3 Determinant of a 3 x 3 matrix by Matrix Enhancement
A (3 x 3) matrix A may be ‘enhanced’ by the repetition of each of four ‘corner’
elements as shown below
From the enhanced matrix we can now define three positive diagonals, shown below by
solid lines and three negative diagonals, shown below by broken lines.
Multiplying along the diagonals, we can calculate the determinant as the sum of the three
positive diagonals less the three negative diagonals. The complete expression is then
a13 a21 a32 + a11 a22 a33 + a12 a23 a31 – a33 a21 a12 – a31 a22 a13 – a32 a23 a11.
The above expression gives Leibniz formula for a (3 x 3) matrix.
Example 1
Calculate the determinant of the matrix by Matrix Enhancement
method.
The enhanced matrix becomes
and its determinant will be
= (2).(1).(-1) +(1).(1).(3)+ (2).(1).(1) - (3).(1).(2) - (1).(1).(2) – (-1).(1).(1)
= -2+3+2-6-2+1
= -4
1.3.4 Leibniz formula to compute determinant of a Matrix
Here the sum is computed over all permutations ó of the numbers {1,2,...,n} and sgn(ó)
denotes the signature of the permutation ó: +1 if ó is an even permutation and - 1 if it is
odd.
This formula contains n! (factorial) summands, and it is therefore impractical to use it to
calculate determinants for large n.
For small matrices, one obtains the following formulas:
31131
111
21221

if A is a 1-by-1 matrix, then
if A is a 2-by-2 matrix, then
for a 3-by-3 matrix A, the formula is more complicated:
which takes the shape of the Sarrus' scheme.
1.3.5 Laplace's formula to compute determinant of a Matrix
It is also possible to expand a determinant along a row or column using Laplace's
formula, which is efficient for relatively small matrices. To do this along row i, say, we
write
where the Ci,j represent the matrix cofactors, i.e. Ci,j is ( - 1)i + j times the minor Mi,j,
which is the determinant of the matrix that results from A by removing the i-th row and
the j-th column.
Example 2
Suppose we want to compute the determinant of
We can go ahead and use the Leibniz formula directly:
Alternatively, we can use Laplace's formula to expand the determinant along a row or
column. It is best to choose a row or column with many zeros, so we will expand along
the second column:
In order to compute the determinant of a given matrix, we are using the Laplace’s
Formula.
1.4 Inverse of a Matrix
1.4.1 Cofactors
The cofactor of the element ai,j equals (-1)i+j det(B) where B is the matrix formed by
deleting row i and column j from A.
Therefore the cofactor of an element of a 3 x 3 determinant is the
2 x 2 determinant obtained by deleting the row and column containing that element and
multiplying by +1 or –1 according to the pattern:
A cofactor is always designated by the capital letter corresponding to the element to
which it belongs.
1.4.2 Adjoint of a Matrix
The standard notation for ‘the adjoint of matrix A’ is adj(A). Notice that adj(A) is the
transpose of the matrix formed by replacing each element of A by its cofactor. The
product A adj(A) is always a diagonal matrix.
where I is the 3 3 unit matrix.
1.4.3 Inverse of a Matrix - Definition
In linear algebra, an n-by-n (square) matrix A is called invertible o r non-singular if
there exists an n-by-n matrix B such that
AB = BA = In
where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix
multiplication. If this is the case, then the matrix B is uniquely determined by A and is
called the inverse of A, denoted by A-1. It follows from the theory of matrices that if
AB = IAAN
for square matrices A a n d B, then also B A =
I.
Matrix inversion is the process of finding the matrix B that satisfies the prior equation for
a given invertible matrix A.
Remark: The symbol A-1 is read “A inverse”.
If a matrix A has an inverse, then A is invertible (or nonsingular); otherwise, A is
singular. A nonsquare matrix cannot have an inverse. Moreover not all square matrices
possess inverse. If, however, a matrix does possess an inverse, then that inverse is unique.
1.4.4 Properties of invertible matrices
Let A be a square n by n matrix over a field K (for example the field R of real
numbers).
Then the following statements are equivalent:
A is invertible.
A is row-equivalent to the n-by-n identity matrix In.
A is column-equivalent to the n-by-n identity matrix In.
A has n pivot positions.
det A 0.
rank A = n.
The equation Ax = 0 has only the trivial solution x = 0 (i.e., Null A = {0})
The equation Ax = b has exactly one solution for each b in Kn.
The columns of A are linearly independent.
The columns of A span Kn (i.e. Col A = Kn).
The columns of A form a basis of Kn.
The linear transformation mapping x to Ax is a bijection from Kn to Kn.
There is an n by n matrix B such that AB = In.
The transpose AT is an invertible matrix.
The matrix times its transpose, AT × A is an invertible matrix.
The number 0 is not an eigenvalue of A.
1.4.5 Matrix inverses in real-time simulations - Applications
Matrix inversion plays a significant role in computer graphics, particularly in 3D
graphics rendering 3D simulations. Examples include screen-to-world ray casting, worldtosubspace-to-world object transformations, and physical simulations. The problem is
usually the numerical complexity of calculating the inverses of 3×3 and 4×4 matrices.
Compared to matrix multiplication or creation of rotation matrices, matrix inversion is
several orders of magnitude slower. There are existing solutions which use hand-crafted
assembly language routines and SIMD processor extensions (SSE, SSE2, Altivec) that
address this problem and which achieve a performance improvement of as much as five
times.
1.4.6 Analytic solution
In other words , the inverse of is adj( )
We know that adj( ) i.e. adj( )
A-1 A A
AAIAAI
Find the cofactor of the elements.
1(2) 0 1( 18) 2 18 20
| A | 1(-28 30) 0( 21 0) ( 1)( 18 0)
Based on the first row
067
345
101
i) Let A
Solution :
.
067
345
101
Solution :
and "0"denotes the zeromatrix and hence find the inverse of A.
satisfies the equation A 4A 5I 0 where I is the identity matrix
221
212
122
Matrix
A matrix (plural matrices) is a rectangular table of elements (or entries).
A matrix having m rows and n columns is said to be of order m n. If m = n, the
matrix is square of order n. For a square matrix, the entries a11, a22, a33,…, ann are the
main diagonals.
A matrix that has only one row is a row matrix, and a matrix that has only one
column is a column matrix
Diagonal Matrix
A square matrix A is said to be a diagonal matrix if aij= 0 when i j. In a diagonal
matrix all the entries except the entries along the main diagonal is zero.
Triangular matrix
A square matrix in which all the entries above the main diagonal are zero is called
a lower triangular matrix. If all the entries below the main diagonal are zero, it is
called an upper triangular matrix.
Scalar Matrix
A scalar matrix is a diagonal matrix in which all the entries along the main
diagonal are equal.
Identity Matrix or Unit Matrix
An Identity Matrix or Unit Matrix is a scalar matrix in which entries along the
main diagonal are equal to 1.
Zero or Null or Void Matrix
In which all the entries are zero.
Equality of matrices
The matrices A=[aij]mxn and B=[bij]pxq are equal if m = p, n = q and aij = bij for
every i and j.
Cofactor
The cofactor of the element ai,j equals (-1)i+j det(B) where B is the matrix formed
by deleting row i and column j from A.
Inverse of matrix
det(A)
A-1 adj(A)
1.7 Lesson End Activities
only one inverse.
7) Define the inverse of a matrix and prove that, if a matrix has an inverse, it has
123
111
111
6) Find the inverse of A
is A itself.
443
101
-4-3-3
is 3A and that the adjoint of A
2-21
21-2
-1 - 2 - 2
a
6) Show that the adjoint of
312
-1 4 5
207
b)
1 5 12
135
123
a)
5) Compute the adjoint of
872
26a
543
4) Find the cofactor of 'a' in the determinant
1cc
1bb
1aa
bc ca ab
abc
abc
3) Without expanding, verify that
832
263
533
2) Evaluate
gfc
hbf
ahg
Rank of a Matrix
Contents:
2.0 Aims and objectives
2.1 Introduction
2.2 Rank of a Matrix
2.2.1 Rank-Definitions
2.2.2 Properties
2.2.3 Computations of the rank – Method 1
2.2.4 Solved Problems
2.3 Elementary row operations
2.3.1 Row-Echelon Form
2.3.2 Computations of the rank – Method 2
2.3.3 Applications of the Rank
2.4 Let Us Sum Up
2.5 Lesson End Activities
2.6 References
2.0 Aims and objectives
In this lesson, we have discussed the rank of a matrix and its computation
methods. Elementary row operations in matrices are also discussed in this lesson.
After reading this lesson, you should able to understand
Rank and its properties
Rank computation methods.
Elementary row operations and Row-Echelon Form.
Applications of calculating the rank.
2.1 Introduction
The maximum number of independent rows (and therefore columns) that can be
found in a matrix is called the rank of a matrix. One useful application of calculating the
rank of a matrix is the computation of the number of solutions of a system of linear
equations. The system is inconsistent or having at least one solution or unique solution
can be found using its rank.
In control theory, the rank of a matrix can be used to determine whether a linear system is
controllable, or observable.
2.2 Rank of a Matrix
2.2.1 Rank-Definitions
Definition 1
If a matrix has more rows than columns it must necessarily have linear dependence in the
rows. Conversely, if it has more columns than rows there will be linear dependence in the
columns. We define the rank of a matrix, r(A), as the maximum number of
independent rows(and therefore columns) that can be found in the matrix. It is
formally, written as follows
Rule: If we have a set of m equations in n unknowns written as Ax = b
there will be a unique solution if and only if r(A)=n.
Rule: A square matrix (A) of dimension n will have an inverse if and only if r(A)=n.
Alternatively the rank of a matrix can be defined as follows.
Definition 2
A matrix A is said to be of rank r when
(i) at least one minor of A of order r is not zero and
(ii) every minor of A of order r+1 is zero.
(or)
The rank of a matrix is the largest order of any non-vanishing minor of the
matrix.
The rank of an m n matrix is at most
min(m,n). A matrix that has as large a rank as possible is said to have full rank;
otherwise, the matrix is rank deficient.
2.2 Properties
only the zero matrix has rank 0
the rank of A is at most min(m,n)
In the case of a square matrix A (i.e., m = n), then A is invertible if and only if A
has rank n (that is, A has full rank).
If B is any n-by-k matrix, then the rank of AB is at most the minimum of the rank
of A and the rank of B.
As an example of the "<" case, consider the product
Both factors have rank 1, but the product has rank 0.
If B is an n-by-k matrix with rank n, then AB has the same rank as A.
If C is an l-by-m matrix with rank m, then CA has the same rank as A.
The rank of A is equal to r if and only if there exists an invertible m-by-m matrix
X and an invertible n-by-n matrix Y such that
where Ir denotes the r-by-r identity matrix.
The rank of a matrix plus the nullity of the matrix equals the number of columns
of the matrix (this is the "rank theorem" or the "rank-nullity theorem").
2.2.3 Computations of the rank – Method 1
[Computations of the rank of a matrix from its determinant value]
Example 1
Determine the rank of the following matrix
=0
But there is at least one non-zero minor of order 2, namely which is = -7.
Hence r(A) =2.
Let A
Other non-vanishing determinants of order 2 might be written but a single one is a
sufficient to establish that the rank is 2.
Example 2.
|A| =1(6-1)-2(4-3)+3(2-9)
= 5 – 2 – 21 = -18 0
It is square matrix with det(A)0. Hence r(A) = 3
669
423
246
Let A
Example 3.
|A| = 0
Also each of the second order minor is also = 0.
But the matrix is non-zero, hence r(A) = 1.
2.2.4 Solved Problems
Hence the rank of the matrix is 3.
10
166
1( 1) 2( 3) 3( 2)
1(24 25) 2(12 15) 3(10 12)
356
245
123
|A|
356
245
123
Let A
Solution :
356
245
123
1) Find the rank of
Therfore the rank of the matrix is not 3.
0
24 14 10
1(8) 2(7) 3(10)
3(8 1 0 6) 2(7 1 0 4) ( 1)(7 6 8 4)
461
780
321
|A|
461
780
321
Let A
Solution :
461
780
321
he rank of the matrix, r(A), is 2.
24 14 10 0.
78
32
78
32
Consider the submatrix
Therfore the rank of the matrix, r(A), is 3.
( 2)( 1) 3( 2) 4( 1) 2 6 4 0
( 2)(1 7 4 2) 3(0 7 1 2) 4(0 4 1 1)
147
012
234
|A|
Consider the second 3 3matrix.
Therfore the rank of the matrix is not 3.
0
24 14 10
1(8) 2(7) 3(10)
1(1 7 4 2) 3(1 7 3 2) 4(1 4 3 1)
347
112
134
|A|
Consider the first 3 3matrix.
fourth column respectively.
by deleting first,second, third and
134
011
213
and
137
012
214
,
147
012
234
,
347
112
134
Four square submatrices of order 3 exist
1347
0112
2134
Let A
Solution :
1347
0112
2134
Eigenvaules and Eigenvectors
Contents:
3.0 Aims and objectives
3.1 Introduction
3.2 Linear Transformation
3.2.1 Characteristic equation
3.2.2 Characteristic Matrix
3.2.3 Characteristic Polynomial
3.3 Eigenvalues and Eigenvector
3.3.1 Eigenvalues - Definition
3.3.2 Properties of eigenvaules
3.3.3 Eigenvectors - Definition
3.3.2 Properties of eigenvectors
3.4 Problems and Solutions
3.5 Let Us Sum Up
3.6 Lesson End Activities
3.7 References
3.0 Aims and objectives
In this lesson, we have discussed about the Linear Transformation, Eigenvalues and
Eigenvectors.
After reading this lesson, you should able to
Perform Linear Transformations.
Find eigenvalues and eigenvectors.
Know about the properties of eigenvalues and eigenvectors.
Understand Cayley-Hamilton theorem and to find the inverse of a matrix.
To solve problems using the properties of the eigenvalues and eigenvectors.
3.1 Introduction
Eigenvalues are a special set of scalars associated with a linear system of
equations (i.e., a matrix equation) that are sometimes also known as characteristic roots,
characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus
and Minc 1988, p. 144).
The determination of the eigenvalues and eigenvectors of a system is extremely
important in physics and engineering, where it is equivalent to matrix diagonalization and
arises in such common applications as stability analysis, the physics of rotating bodies,
and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired
with a corresponding so-called eigenvector.
3.2 Linear Transformation
Consider a system of n linear equations in n variables.
------------------- > (1)
The above set of equations is said to form a linear transformation i.e. the
equations (1) transform the n variables x1,x2,x3, …, xn into n variables y1,y2 ,…, yn.
Rewriting (1) as Y = A X ------------------- > (2)
Where
The equation (2) is said to be a linear transformation of vector X into another vector Y; A
is called the matrix of transformation and |A| is called the modulus of the transformation.
If |A| = 0 the transformation is singular; otherwise non-singular.
3.2.1 Characteristic equation
The characteristic equation of the matrix A is |A-I| = 0. It is a polynomial equation in .
A matrix A satisfies its own characteristic equation (Cayley-Hamilton theorem)
3.2.2 Characteristic Matrix
The characteristic matrix of A is (A-I) and is a function of the scalar .
3.2.3 Characteristic Polynomial
The characteristic polynomial, p(t), of a matrix A is p(t) = |A-I|.
The characteristic polynomial of A nn is of the form: n -tr(A)* n-1 + ... + -1n |A|.
The characteristic polynomial of A 22 is 2 - tr(A)* + |A|
3.3 Eigenvalues and Eigenvector
3.3.1 Eigenvalues - Definition
The eigenvalues of a matrix are the roots of its characteristic equation. They may
also be referred to by any of the fourteen other combinations of: [characteristic, eigen,
latent, proper, secular] + [number, root, value].
That is, the eigenvalues of matrix A are the roots of its characteristic equation: |A-I| = 0.
The function eig(A) denotes a column vector containing all the eigenvalues of A with
appropriate multiplicities.
3.3.2 Properties of eigenvaules
The matrix A has exactly n eigenvalues (not necessarily distinct)
[Complex]: tr(A) = sum(eig(A))
[Complex]: det(A) = prod(eig(A))
[A:m*m, C:n*n]: eig([A B; 0 C]) = [eig(A); eig(C)]
det(A)=0 iff 0 is an eigenvalue of A
The eigenvalues of a triangular or diagonal matrix are its diagonal elements.
[ Hermitian]: The eigenvalues of A are all real.
[ Unitary]: The eigenvalues of A have unit modulus.
[ Nilpotent]: The eigenvalues of A are all zero.
[ Idempotent]: The eigenvalues of A are all either 0 or 1.
The eigenvalues of Ak are (eig(A))k
Similar matrices have the same eigenvalues
Sum of the eigen values is equal to the sum of the diagonal elements.
The product of the eigen values of a matrix A is equal to its determinant
value.
A square matrix A and its transpose AT have the same eigen values.
If 1, 2, ….,n, are eigen values of matrix A, then
3.3.3 Eigenvectors - Definition
is an eigenvalue of Ann iff for some non-zero x, A x =x. x is then called
an eigenvector corresponding to .
.
3.4 Problems and Solutions
Non-symmetric matrices with non repeated eigenvalues
1) Find the eigenvalues and eigenvectors of the matrix
Solution
The characteristic equation is | A - I | = 0
(4 – ) (2 – ) – 3 = 0
2 – 6+5 = 0
(– 1) (– 5) = 0
= 1 or 5.
The eigenvalues are 1 and 5.
Eigenvectors: When =1, the equation
4-1 1
=
3 2-1
3x1 + x2 = 0
3x1 + x2 = 0
(or) x2 = –3x1
Since the value of x2 is –3 times of x1, the general eigenvector corresponding
to = 1 is w h e r e k i s a c o n s t ant. For different values of k we get different
eigenvectors. Therefore,
for k = 1 the eigenvector is
*Non - symmetricmatrices with non - repeated eigen values.
6) Find the eigen values and eigen vectors of the following matrices.
02
13
5) Find the eigen values and eigen vectors of A
4) State Cayley - Hamilton theorem.
3) Define characteristic equation, eigen values and eigen vectors of a squarematrix A.
(Use property).
005
020
314
2) Find the eigen values of the matrix A
(Use property).
367
234
Unit II
Set Theory
Contents:
4.0 Aims and objectives
4.1 Introduction
4.2 Set and its elements
4.3 Set Description
4.4 Types of Sets
4.5 Venn-Euler Diagrams
4.6 Let Us Sum Up
4.7 Lesson End Activities
4.8 References
4.0 Aims and objectives
In this lesson, we have discussed about set theory and Venn-Euler Diagrams.
After reading this lesson, you should able to
About set and methods of description.
Know about the types of sets.
Represent sets using Venn-Euler Diagrams and to solve problems.
4.1 Introduction
The concept of set theory, originated in 1895 by the German mathematician
G.Cantor, is used in various disciplines. This chapter introduces the notation and
terminology of set theory.
, where p and q are integers and q 0}
4.2 Set and its elements
Definition: A set is a collection of objects.
Examples:
i) The odd integers between 10 and 20.
ii) The vowels in English alphabet
iii) Rivers in India
iv) The planets in the Solar system.
Elements of a set:
The objects of a set are called its elements or members conventionally, capital
letters A, B, C, D, etc. are used to denote sets and lower case letters a, b, c, d, etc. are
used to denote its members.
The symbol is used to indicate ‘belongs to’. The statement ‘p is an element of
A’ is written as p A. The symbol is used to indicate ‘does not belongs to’. The
statement ‘q is not an element of A’ is written as q A.
Some standard sets
Some sets occur very often in the text and so we use special symbols for them. They are
i) The set of Natural numbers i.e. N = { 1, 2, 3, 4, …}
ii) The set of Integers (or countable numbers) i.e. I = { …,-3, -2, -1, 0, 1, 2, 3, …}
iii) The set of Positive Integers i.e. I+ = { 1, 2, 3, …}
iv) The set of rational numbers
Remark
1) The quantitative attributes like honest persons, rich person, beautiful women,
etc do not form sets.
4.3 Set Description
There are two different ways for describing a set. They are
1) Roster method and
2) Set Builder method.
1) Roster Method
In this method the members are represented as a list. For example,
V = {a, e, i, o, u}
denotes the set V whose elements are the letters a, e, i, o, u. Note that the elements
of the set are separated by commas and enclosed in braces {}.
2) Set Builder method
Sometimes we cannot list the elements of a set explicitly. In this method, the
set is defined by stating the properties which characterizing the members. For
example,
V = {x : x is a vowel in English alphabet}
B = {x : x is an even integer, x > 0 }
4.4 Types of Sets
Finite set
A set having finite number of elements or members is known as finite set.
Examples:
1) The set of months in a year.
2) The set of vowels in the English alphabet.
Infinite set
A set having infinite number of elements or members is known as infinite set.
Examples:
3) The set of all natural numbers i.e. N = { 1, 2, 3, 4, …}
4) The set of integers, I = { …, -3, -2, -1, 0, 1, 2, 3, …}
Null set
A set with no elements is called the empty set or null set and is denoted by 
(read as phi) or {}.
For example, A = {x : x3 + 4 = 0, x is real }
Equality of sets
Two sets A and B are said to be equal if every element of A is an element of B
and also every element of B is an element of A. If the sets A and B are equal, they are
denoted by A = B.
For example,
i) If A = {4, 3, 2, 1} and B = {1, 3, 4, 2} then A=B, because both have same
and equal number of elements.
ii) If A = set of all integers whose square is 9,
B = set of all roots of the equation, x2 – 9 = 0 and
C = {-3, +3}
then A = B.
Equivalent Sets
Let A and B are two sets. The sets A and B are said to be equivalent sets if and
only if there exists a one-to-one correspondence between their elements. By one-toone
correspondence we mean that for each element in A, there exists and match with
one element in B and vice versa.
The symbol , ~ or is used to denote equivalent sets.
Illustration
A = {x : x is a letter in the word BOAT}
B = {x : x is a letter in the word CART}
Thus A B.
Subset
Let A and B be two non-empty sets. If every element of A is also an element of B,
then A is called a subset of B. This relationship is written as A B or B A.
If A is not a subset of B, i.e. if at least one element of A does not belong to B, we
write A B.
Theorem
i) For any set A, we have A U.
ii) For every set A, we have A A.
iii) If A B and B C, then A C.
iv) A = B if and only if A B and B A.
If A B, then it is still possible that A = B.
When A B but AB, we say A as a proper subset of B. For example, suppose
A = {1,3} B = {1, 2, 3} C = {1, 3, 2}. Then both A and B are subsets of C; but A
is a proper subset of C, where as B is not a proper subset of C since B = C.
Power Set
If A is any set, then the family of all the subsets of A is called power set of A and
is denoted by P(A), i.e. P(A) = { B : BA}. Obviously, and A are both members of
P(A).
Illustrations
i) Let A = {b}, then P(A) = { , {b}}
ii) Let A = , then P(A) = {}
iii) If A = {8, 9}, then P(A) = { , {8}, {9},{8,9}}
Number of subsets
The number of subsets of a set with n elements is 2n -2 (leaving Ø and the full set)
Ø,A are called improper subsets of A.
Number of proper subsets
The number of proper subsets of a set with n elements is 2n – 1.
Universal set
In every problem there is either a stated or implied universe of discourse. The
universe of discourse includes all things under discussion at a given time. In the
mathematical theory of sets, the universe of discourse is called the universal set. The
letter U is typically used for the universal set. The universal set might well change from
problem to problem.
4.5 Venn-Euler Diagrams
In most areas of mathematics, our reasoning can be aided and clarified by
utilizing various kinds of drawings and diagrams. In set theory, we commonly use Venn
diagrams, developed by the logician John Venn (1834-1923). In these diagrams, the
universal set is represented by a rectangle, and other sets of interest within the universal
set are depicted by oval regions, or sometimes by circles or other shapes.
If the sets A and B are equal, then the same circle represents both A and B. If the
sets A and B are disjoint, i.e. they have no elements in common, then circles representing
A and B are drawn in such a way that they have no common area as shown in figure(a).
However if few elements are common both in A and B, then sets A and B are in general
represented as in figure(b).
4.6 Let Us Sum Up
Set
A set is an unordered collection of distinct and distinguishable objects.
Roster Method
In this method the members are represented as a list.
Set Builder method
The set is defined by stating the properties which characterizing the members.
Finite set
A set having finite number of elements or members is known as finite set.
Infinite set
A set having infinite number of elements or members is known as infinite set.
Null set
A set with no elements is called the empty set or null set and is denoted by .
Equality of sets
If every element of A is an element of B and also every element of B is an
element of A then the sets A and B are equal, denoted by A = B.
Equivalent Sets
The sets A and B are said to be equivalent sets if and only if there exists a onetoone correspondence between their elements.
The symbol , ~ or is used to denote equivalent sets.
Subset
If every element of A is also an element of B, then A is called a subset of B. This
relationship is written as A B or B A.
If A is not a subset of B, i.e. if at least one element of A does not belong to B, we
write A B.
If A is any set, then the family of all the subset of A is called power set of A
and is denoted by P(A), i.e. P(A) = { B : BA}. Obviously, and A are both
members of P(A).
Number of subsets
The number of subsets of a set with n elements is 2n.
Number of proper subsets
The number of proper subsets of a set with n elements is 2n – 2.
Universal set
The universe of discourse is called the universal set. The letter U is used for the
universal set.
Venn-Euler Diagrams
In these diagrams, the universal set is represented by a rectangle, and other sets of
interest within the universal set are depicted by oval regions, or sometimes by circles
or other shapes.
4.7 Lesson End Activities
1. List the members of these sets.
a) {x | x is a real number such that x2 = 1}
b) {x | x is a positive integer less than 12}
c) {x | x is the square of an integer and x < 100}
d) {x | x is an integer such that x2 — 2}
2. Use set builder notation to give a description of each of these sets.
a) {0,3,6,9, 12}
b) {-3,-2,-1,0,1,2,3}
c) {m, n, o, p}
3. Determine whether each of these pairs of sets are equal.
a) {1,3,3,3,5,5,5,5,5}, {5,3, 1}
b) {{1}}, {1,{1}}
c) ,{}
4. Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D ={4, 6, 8}.
Determine which of these sets are subsets of which other of these sets.
5. For each of the following sets, determine whether 2 is an element of that set.
a) {x R | x is an integer greater than 1}
b) {x R | x is the square of an integer}
c) {2,{2}}
d) {{2},{{2}}}
e) {{ 2 }, {2, { 2 } } }
f) {{{2}}}
6. For each of the sets in Exercise 5, determine whether {2} is an element of that
set.
7. Use a Venn diagram to illustrate the relationship A B and B C.
8. Suppose that A, B, and C are sets such that A B and B C. Show that A C.
9. Find two sets A and B such that A B and A B.
10. What is the cardinality of each of these sets?
a) {a} b) {{a}}
c) {a,{a}} d) {a,{a},{a,{a}}}
11.What is the cardinality of each of these sets?
a) b) {}
c) {, {}} d) {,{},{,{}}}
12 Find the power set of each of these sets.
a) {a) b) {a,b} c) {, {}}
13. Can you conclude that A = B if A and B are two sets with the same power set?
14. How many elements does each of these sets have?
a) P({a, b,{a, b}})
b) P({8,a,{a},{{a}}})
c) P(P())
4.8 References
1) J K Sharma, ”Discrete Mathematics”
2) Kenneth H.Rosen, “Discrete Mathematics and its Applications”
3) Seymour Lipschutz and Marc Lipson, “Discrete Mathematics”
4) Charles D.Miller and Others, “Mathematical Ideas”
5) Wikipedia, the free encyclopedia.
Set operations and laws of set theory
Contents:
5.0 Aims and objectives
5.1 Introduction
5.2 Set operations and laws of set theory.
5.2.1 Union of sets
5.2.2 Intersection
5.2.3 Disjoint
5.2.4 Difference of two sets
5.2.5 Complement of a set
5.3 Algebra of sets and Duality
5.3.1 Set Identities
5.3.2 Duality
5.3.3 Proofs using set builder notation, membership tables and Venn diagrams
5.4 Partitions of sets
5.5 Minsets
5.6 Let Us Sum Up
5.7 Lesson End Activities
5.0 Aims and objectives
In this lesson, we have discussed about set operations, laws of set theory, algebra of
sets and duality.
After reading this lesson, you should able to
Do set operations like Union, Intersection, and Disjoint.
Find the set differences and set complements.
Know about the set identities and different methods to prove them.
Know about the duality and finding the duality of an equation.
Partition the sets and to generate minsets.
5.1 Introduction
In this section we will discuss the various operators that are used to combine two
or more sets and how to work with sets using set algebra and duality.
.
5.2 Set operations and laws of set theory.
Two or more sets can be combined in many different ways. In this section we will
discuss the various operators that are used for this purpose.
5.2.1 Union of sets
Let A and B be sets. The union of A and B, denoted by A B, is the set that
contains those elements that are either in A or in B, or in both.
An element x belongs to the union of the sets A and B if and only if x belongs to
A or x belongs to B. Symbolically,
A B = { x : x A or x B }
Illustration
If A = {1,2,3,4} and B={2,8,9}, then A B = {1,2,3,4,8,9}
5.2.2 Intersection
Let A and B be sets. The intersection of A and B, denoted by A B, is the set that
containing those elements in both A and B.
An element x belongs to the intersection of the sets A and B if and only if x
belongs to A and x belongs to B. Symbolically,
A B = {x : x A and x B }
5.2.3 Disjoint
Let A and B be sets. If intersection of the sets A and B is an empty set, then these
two sets are said to be disjoint. Symbolically,
A B = 
Example 1
Let A = {1,3,5,7,9} and B = {2,4,6,8,10}. Since A B = , A and B are disjoint.
5.2.4 Difference of two sets
Let A and B be sets. The difference of A and B denoted by A – B or A\B, is the
set having those elements that are in A but not in B.
An element x belongs to the difference of the sets A and B if and only if x belongs
to A and x does not belong to B. Symbolically,
A\B = A – B = { x : x A and x B }
Properties of Difference of two sets
1) A – A = 
2) A – = A
3) A – B A
4) A – B, A B and B-A are mutually disjoint
5) (A – B) B = 
6) (A – B) A = A
5.2.5 Complement of a set
Let U be the universal set and A be any set. The complement of A, denoted by A’,
is the complement of A with respect to U.
An element x belongs to A’ if and only if x A. This tells us that
A’ = {x : x A}
In fig. 5.1 the shaded area outside the circle representing A is the area
representing A’.
Example 2
1. If N = { 1,2,3,4,…} is the universal set and
let A = {1,3,5,7, …}, then A’ = N – A = { 2,4,6,8,…}
2. U’ = and ’ = U
Properties
1. A A’ = U
2. A A’ = 
3. U’ = 
4. ’ = U
5. (A’)’ = A
6. (A – B) = A B’
7. If A B, then A (B – A) = B
8. (A B)’ = A’ B’
9. (A B)’ = A’ B’
5.3 Algebra of sets and Duality
5.3.1 Set Identities
Sets under the operation of Union, Intersection, and Complement satisfy various
laws or identities. The table 5.1 lists the most important set identities. These identities can
be proved by different methods. We will prove some of these identities in this lesson and
the proof of the remaining identities will be left as exercises.
Table: 5.1 Set Identities
Identity Name
A = A
A U = A
Identity laws
A U = U
A = 
Domination laws
A A = A
A A = A
Idempotent laws
(A’)’ = A Complementation law
A B = B A
A B = B A
Commutative laws
A (B C) = (A B) C
A (B C) = (A B) C
Associative laws
A (B C) = (A B) (A C)
A (B C) = (A B) (A C)
Distributive laws
(A B)’ = A’ B’
(A B)’ = A’ B’
De Morgan’s laws
A (A B) = A
A (A B) = A
Absorption laws
A A’ = U
A A’ = 
Complement laws
5.3.2 Duality
Note that the identities in table 5.1 are arranged in pairs. The principle behind this
arrangement is just the replacement of sets and operators. Suppose E is an equation of set
algebra. The dual E* of E is the equation obtained by replacing each occurrence of , ,
U and in E by , , and U, respectively. For example, the dual of
(U A) (B A) = A is (A) (B A) = A
5.3.3 Proofs using set builder notation, membership tables and Venn diagrams
a) Prove the distributive laws using set builder notation and logical equivalence.
1) Prove A (B C) = (A B) (A C)
Proof: Let x is an arbitrary element of A È (B È C). Then
x A (B C) x A or x (B C)
x A or (x B and x C)
(x A or x B) and (x A or x C)
x (A B) and x (A C)
x (A B) (A C)
Hence A (B C) = (A B) (A C)
2) Prove A (B C) = (A B) (A C)
Proof: Let x is an arbitrary element of A (B C). Then
x A (B C) x A and x (B C)
x A and (x B or x C)
(x A and x B) or (x A and x C)
x (A B) and x (A C)
x (A B) (A C)
Hence A (B C) = (A B) (A C)
b) Prove the distributive laws using membership tables.
Set identities can also be proved using membership tables. To indicate that an
element in a set, a Y is used; to indicate that an element is not in a set, a N is used.
Table: 1
A membership table for the Distributive law A (B C) = (A B) (A C)
A B C B C A (B C) A B A C (A B ) (A C)
YYYYYYYY
YYNYYYNY
YNYYYNYY
YNNNNNNN
NYYYNNNN
NYNYNNNN
NNYYNNNN
NNNNNNNN
Table: 2
A membership table for the Distributive law A (B C) = (A B) (A C)
A B C B C A (B C) A B A C (A B ) (A C)
YYYYYYYY
YYNNYYYY
YNYNYYYY
YNNNYYYY
NYYYYYYY
NYNNNYNN
NNYNNNYN
NNNNNNNN
c) Prove the Distributive law using Venn diagrams
Solution:
A (B C) = (A B) (A C)
Venn diagrams for LHS:
Venn diagrams for RHS:
Fig. 5.2
Here the figures 5.2 (b) and 5.2 (e) are same and hence proved.
5.4 Partitions of sets
In order to analyze a bigger set easily, we can be split them into smaller, nonoverlapping
and non-empty, subsets. For example, students in a class may be partitioned
into small groups on the basis of certain criteria so as to keep track of their progress and
regular internal assessment.
Let A be a non-empty set. The partition of A is any set of non-empty, nonoverlapping
subsets A1, A2, A3, A4,…, An such that
i) A = A1 A2 … An
ii) The subsets Ai are mutually disjoint, i.e. Ai Aj = for i = j.
Figure Fig .5.3 is the Venn diagram showing partition of set A into five subset A1, A2, A3,
A4 and A5.
Example 3
Let A = {10,11,12,13}. Then A1 ={10}, A2 = {11,12}, A3 ={13} are the partitions
of A.
5.5 Minsets
Let X = {B1,B2,B3,…,Bn} where the elements B1,B2, etc are the subsets of a set A.
Then the set of the form C1 C2 C3 … Cn, where each Ci may be either Bi or its
complement i.e. Bi
’ is called a minset or minterm, generated by Bi s (i = 1,2,...,n).
Example 4
Let B1 = {1,4,6} and B2 = {1,3,4} be two subsets of A = {1,2,3,4,5,6}. In order to
have a partition of A without repetition of elements, we may describe B1 and B2 as
follows:
C B B' {1,4,6} {2,5,6} {6}
1 1 2 
C B B {1,4,6} {1,3,4} {1,4} 2 1 2 
C B B {1,4,6} {2,5,6} {3} 2
'1

C B B' {2,3,4} {2,5,6} {2,5}
3
2
'1

As shown in fig. 5.4, none of the sets C1,C2,C3 and C4 have elements in common, the set
generated by B1 and B2 is the partition of A.
5.6 Let Us Sum Up
Union
The union of the sets A and B, denoted by A B, is the set that contains those
elements that are either in A or in B, or in both.
4
Intersection
The intersection of sets A and B, denoted by A B, is the set that contains those
elements in both A and B.
Disjoint
If intersection of the sets A and B is an empty set, then these two sets are said to
be disjoint. Symbolically, A B = 
Difference of two sets
The difference of A and B denoted by A – B or A\B, is the set having those
elements that are in A but not in B.
Properties of Difference of two sets
A – A = 
A – = A
A – B A
A – B, A B and B-A are mutually disjoint
(A – B) A B = 
(A – B) A – A
Complement of a set
Let U be the universal set and A be any set. The complement of A, denoted by A’,
is the complement of A with respect to U.
Properties
A A’ = U
A A’ = 
U’ = 
’ = U
(A’)’ = A
(A – B) = A B’
If A B, then A (B – A) = B
(A B)’ = A’ B’
(A B)’ = A’ B’
Set Identities
Set Identities
Identity Name
A = A
A U = A
Identity laws
A U = U
A = 
Domination laws
A A = A
A A = A
Idempotent laws
(A’)’ = A Complementation law
A B = B A
A B = B A
Commutative laws
A (B C) = (A B) C
A (B C) = (A B) C
Associative laws
A (B C) = (A B) (A C)
A (B C) = (A B) (A C)
Distributive laws
(A B)’ = A’ B’
(A B)’ = A’ B’
De Morgan’s laws
A (A B) = A
A (A B) = A
Absorption laws
A A’ = U
A A’ = 
Complement laws
Duality
The dual E* of E is the equation obtained by replacing each occurrence of , , U
and in E by , , and U, respectively.
Partitions of sets
Let A be a non-empty set. The partition of A is any set of non-empty, nonoverlapping
subsets A1, A2, A3, A4,…, An such that
i) A = A1 A2 … An
ii) The subsets Ai are mutually disjoint, i.e. Ai Aj = for i = j.
5.7 Lesson End Activities
1. Let A = {1.2,3.4.5} and B = {0, 3, 6}. Find
a) A B b) A B
c) A – B d) B – A.
2. Let A={a, b, c, d, e} and B ={a, b, c, d, e, f, g, h}. Find
a) A B b) A B
c) A – B d) B – A.
3. Let A be a set. Show that (A’)’ = A.
4. Let A be a set. Show that
a) A = A. b) A = .
c) A A = A. d) A A = A.
e) A - = A. f) AU =U.
g) A U = A. h) - A = .
5. Let A and B be sets. Show that
a) A B = B A. b) A B = B A.
6. Show that if A and B are sets, then A –B = A B’.
7. Show that if A and B are sets, then (A B) ( A B’) = A.
8. Let A, B, and C be sets. Show that
a) A(BC) = (AB)C.
b) A(B C) = (AB)C.
c) A(BC) = (AB)(AC).
9. Let A, B, and C be sets. Show that (A – B) – C = (A – C) – (B – C).
10. Draw the Venn diagrams for each of these combinations of the sets A, B, and C.
a) A (B C)
b) A’ B’ C’
c) (A – B) (A –C) (B – C)
The inclusion-Exclusion Principle
Contents:
6.0 Aims and objectives
6.1 Introduction
6.2 The inclusion-Exclusion Principle
6.3 Problems and Solutions
6.4 Lesson End Activities
6.5 References
6.0 Aims and objectives
In this lesson, we have discussed about the Inclusion-Exclusion Principle.
After reading this lesson, you should be able to
Know what do you mean by an Inclusion-Exclusion Principle.
Apply Inclusion-Exclusion Principle to solve problems.
6.1 Introduction
The Inclusion-Exclusion Principle method is used in finding the number of
elements in a finite set (also called cardinal number). While applying this principle some
terms are Included and some terms are excluded from an expression, so the name
Inclusion-Exclusion Principle.
6.2 The inclusion-Exclusion Principle
Let A and B be any finite sets. Then
n(A B) = n(A) + n(B) - n(A B)
In other words, to find the number n(A B) of elements in the union, we add n(A) and
n(B) and then we subtract n(A B); that is, we “Include” n(A) and n(B), and we
“exclude” n(A B).This method of finding the number of elements in a finite set ( also
called cardinal number) is known as Inclusion-Exclusion principle. The cardinal number
of a set A is denoted by n(A). This principle holds for any number of sets like
n(ABC) and so on.
Theorem 1: For any finite sets A, B, C we have
n(ABC) = n(A) + n(B) + n(C) – n(AB) – n(AC) – n(BC) + n(ABC)
Theorem 2: Let A1, A2, … , An be finite sets. Then
Example 1
i) Let A = {1,2,3,4}, then n(A) = 4 since A is having only four elements.
ii) Let A = {1,2,3,4} and B = {4,6,8,9} then A B = {1,2,3,4,6,8,9} and
A B={4}. Therefore n(A) = 4, n(B) = 4 ,n(AB)=7 and n(AB)=1.
The above can be verified by using the Inclusion-Exclusion principle
n(AB)=n(A) + n(B) – n(AB).
By applying the principle we get n(AB)= 4+4–1=7.
Results
Simple but important results on the cardinality of sets are
1) n(A B) n(A) + n(B) when A B = 
2) n(A B) Min {n(A), n(B)}
3) n(A B) = n(A) + n(B) - 2n (A B)
4) n(A - B) n(A) - n(B)
5) n(A B) = n(A) + n(B) - n(A B)
6) (a) n(ABC) = n(A) + n(B)+n(C) - n(AB) - n(BC) - n(A C) + n(ABC)
(b) If the sets A, Band C are mutually disjoint, then
n(A B C) = n(A) + n(B) + n(C)
7) n(A’) = n(U) - n(A)
8) (a) n(A) = n[(A B’) (A B)] = n[(A - B) (A B)] = n(A - B) + n(A B)
(b) n(B) = n(B - A) + n(A B)
9) n(A B) = n(A - B) + n(A B) + n(B - A)
6.3 Problems and Solutions
1) In a group of 200 people, each of whom is at least accountant or management
consultant or sales manager, it was found that 80 are accountants, 110 are management
consultants and 130 are sales managers, 25are accountants as well as sales managers, 70
are management consultants as well as sales managers, 10 are accountants, management
consultants as well as sales managers. Find the number of those people who are
accountants as well as management consultants but not sales managers.
Solution Suppose A, M and S denote the set of accountants, management consultants and
sales managers, respectively. Then it is given that
n(A) = 80, n(M) = 110, n(S) = 130, n(A S) = 25
n(M S) = 70 and n(A M S) = 10
Now, find the number of those people who are accountants as well as
management consultants but not sales managers, i.e. n (A M S’).
n(A M S) = n(A) + n(M) + n(S)-n(A M)-n(M S)-n(A S) + n(A M S)
200 = 80 + 110 + 130 - n(A M) - 70 - 25 + 10
200 = 330 - 95 - n (A M)
n(A M) = 330 - 200 - 95 = 35
Now n(A M) = n(A M S’) + n(A M S)
35 = n(A M S’) + 10
or n(A M S’)= 35 - 10 = 25
2) A TV survey shows that 60 per cent people see program A, 50 per cent see program B,
50 per cent see program C, 30 per cent see program A and B, 20 per cent see program B
and C, 30 per cent see program A and C, and 10 per cent do not see any program. Find
(a) What per cent see program A, Band C? .
(b) What per cent see exactly two programs?
(c) What per cent see program A only?
Solution Suppose X, Y and Z denote the set of people who see program A, Band C,
respectively. Then it is given that
n(X) = 60, n(Y) = 50, n(Z) = 50, n(X Y) = 30, n(Y Z) = 20, n(X Z) = 30
n[(X Y Z)’] = 10 .
(a) Let n( X Y Z) + n[ (X Y Z)’] = 100.
Then n( X Y Z ) = 100 - n[ (X Y Z)’] = 100 - 10 = 90
But n( X Y Z) = n(X)+n(Y)+n(Z)-n(X Y)-n(Y Z)- n(X Z)+n( XYZ)
90 = 60 + 50 + 50 - 30 - 20 - 30 + n(X Y Z)
or n(X n Y n Z) = 90 - 80 = 10
Thus, 10 per cent people see program A, Band C.
(b) Since the set of people who see program A and B but not C, i.e. X Y Z’ and the
set of people who see all the program A, Band C, i.e. X Y Z are disjoint sets,
therefore
n( X Y Z’) + n( X Y Z) = n( X Y )
n(X Y Z’) = n(X Y) - n(X Y Z) = 30 - 10 = 20,
n(X Y’ Z) + n(X Y Z) = n(X n Z)
n(Z Y’ Z) = n(X Z) - n(X Y Z) = 30 - 10 = 20,
n(XC n Y n Z) + n(X n.Y n Z) = n(Y n Z)
n(X’ Y Z) = n(Y Z) - n(X Y Z) = 20 - 10 = 10.
Thus, the percentage of people who see exactly two program = 20 + 20 + 10 = 50.
(b) Since n(X Y’ Z’) = n(X) –n(X Y) - n(X Z) + n(X Y Z)
= 60 - 30 - 30 + 10 = 10
Thus, the percentage of people who see program A only is 10.
3) In a pollution study of 1,500 Indian rivers, the following data were reported: 520 were
polluted by sulphur compounds, 335 were polluted by phosphates, 425 were polluted by
crude oil, 100 were polluted by both crude oil and sulphur compounds, 180 were polluted
by sulphur compounds and phosphates, 150 were polluted by both phosphates and crude
oil and 28 were polluted by sulphur compounds, phosphates and crude oil. How many of
the rivers were polluted by at least one of the three impurities? How many rivers were not
polluted by exactly one of the three impurities? How many of the rivers were not
polluted?
Solution: Suppose S, P and C denote the set of rivers polluted by sulphur compounds,
phosphates and crude oil, respectively. Then it is given that .
n(S) = 520, n(P) = 335, n(C) = 425, n(C S) = 100,
n(S P) = 180, n(P C) = 150, n(S P C) = 28.
(a) Number of rivers polluted by at least One of the three impurities
n(S P C) = n(S)+n(P) + n(C)-n(S P)-n(PC)-n(CS) + n(SPC)
= 520 + 335 + 425 - 100 - 180 - 150 + 28 = 878
(b) Number of rivers polluted by exactly sulphur compounds
n(S P’C’) = n[ S (P C)’] = n(S) - n(S P) - n(S C) + n(S P C)
= 520 - 180 - 100 + 28 = 268
(c) Number of rivers polluted by exactly phosphate
n(P S’ C’) = n[P (S C)’] = n(P) - n(P S) - n(P C) + n(P S C)
= 335-180-150+28 =33
(d) Number of rivers polluted by exactly crude oil
n(C P’ S’) = n[C (P’ S’)] = n(C) - n(C P) - n(C S) + n(C S P)
= 425 - 150 - 100 + 28 = 203.
Thus, the number of rivers which were polluted by exactly one of the three impurities
=268 + 33 + 203 = 504
and so the number of rivers which were not polluted at all
= 1500 - 872 = 622.
4) Of a group of 20 persons, 10 are interested in music, 7 are interested in photography,
and 4 like swimming; further more 4 are interested in both music and photography, 3 are
interested in both music and swimming, 2 are interested in both photography and
swimming and one is interested in music, photography and swimming. How many are
interested in photography but not in music and swimming?
Solution Suppose M, P and S denote the set of persons interested in music, photography
and swimming, respectively. Then it is given that
n(M) = 10, . n(P) = 7, n(S) = 4, n(M P) = 4, n(M S) = 3
n(P S) = 2 and n(M P S)=1.
Number of persons interested in photography but not in music and swimming is
given by
n(P M’ S’)= n[P (M S)’] = n(P) - n(P M) - n(P S) + n(PM S)
=7–4–2+1=2
5) Out of 450 students in a school, 193 students read Science and 200 students read
Commerce, 80 students read neither. Find out how many read both.
Solution Suppose A and B denote the' set of students who read Science and Commerce,
respectively.
It is given that . . ;
n(A) = 193, n(B) = 200, n(U)= 450 and n(A’ B’) = 80.
Now find the number of those students who read Science as well as Commerce, i.e.
n(A n B).
Since A’ B’ = (A B)’, therefore n(A B)’ = 80. But
n(A u B)’ = n(U) - n(A B)
80 = 450 - n(A u B) or n(A u B) = 450 - 80 = 370
Further, we know that n(A B) = n(A) + n(B) - n(A B)
370 = 193 + 200 - n(A B)
n(A B) = 23
Thus, 23 students read both science and commerce.
6) In a survey concerning the smoking habits of consumers, it was found that 55 per cent
smoke cigarette A, 50 per cent smoke B, 42 per cent smoke C, 28 per cent smoke A and
B, 20 per cent smoke A and C, 12 per cent smoke Band C and 10 per cent smoke all the
three cigarettes.
(a) What percentage does not smoke?
(b) What percentage smokes exactly two brands of cigarettes?
Solution Let A, B, C be the sets of persons who smoke brand A, B, and C, respectively.
It is given that
n(A) = 55; n(B) = 50; n(C) = 42; n(A B) = 28;
n(A C) = 20; n(B C) = 12; and n(A B C) = 10.
(a) Now (A B C) is the set of all person who smoke either A or B or C; or any two
brands or all the brands and (A u B u C)’ is the set of persons who do not smoke.
Therefore,
n(A B C) = n(A) + n(B) + n(C)-n(A B)- (B C)-n(A C)+n(A B C)
= 55 + 50 + 42 – 28 –12 –20 + 10 = 97
But, n(A B C)’= 100 - n(A B C) = 100 - 97 = 3
Hence, 3 per cent persons do not smoke.
(b) A B C’ is the set of persons who smoke A and B, but not C and (A n B n C) is
the set of persons who smoke all the 3 brands A, B, and C. Therefore
n(A B) = n(A B C’) + n(A B C)
28 = n(A B C’) + 10 or n(A B C’) = 18.
Similarly, n(A B’ C) = 10 and n(A’ B C) = 2.
Hence, total required number = 18 + 10 + 2 = 30.
7) In a class of 25 students, 12 have taken economics, 8 have taken economics but not
political science. Find the number of students who have taken economics and political
science and those who have taken politics but not economics.
Solution Suppose A and B denote the set of students who take economics and political
science, respectively. It is given that
n(A) = 12, n(A u B) = 25, and n(A n BC) = 8.
Now we have to find the number of students who have taken economics and
political science, i.e. n(A B) and the number of students who have taken political
science but not economics, i.e. n(B A’). Now
n(A) = n(A B’) + n(A B)
or 12 = 8 + n(A B) or n(A B) = 12 – 8 = 4
Also n(A B) = n(A) + n(B) – n(A B)
25 = 12 + n(B) – 4 or n(B) = 17.
Again, n(B) = n(A B) + n(B A’).
Therefore
17 = 4+ n(B A’) or n(B A’)= 17 – 4 = 13.
8) Determine the cardinalities of the sets
(a) A = {n7 : n is a positive integer} (b) B = {n109 : n is a positive integer} (c) A B.
Solution
(a) The cardinality of the set A is infinity because its elements, i.e., number of
positive integers which it contains are infinite.
(b) The cardinality of the set is also infinite because of the same reason as in (i).
(c) Since both A and B are infinite sets,. therefore the cardinality of their union is
also infinite.
6.4 Let Us Sum Up
Theorems
1) For any finite sets A, B, C we have
n(ABC) = n(A) + n(B) + n(C) – n(AB) – n(AC) – n(BC) + n(ABC)
2) Let A1, A2, … , An be finite sets. Then
Results
1. n(A B) n(A) + n(B) when A B = 
2. n(A B) Min {n(A), n(B)}
3. n(A B) = n(A) + n(B) - 2n (A B)
4. n(A - B) n(A) - n(B)
5. n(A B) = n(A) + n(B) - n(A B)
6. (a) n(ABC) = n(A) + n(B)+n(C) - n(AB) - n(BC) - n(A C) + n(ABC)
(b) If the sets A, Band C are mutually disjoint, then
n(A B C) = n(A) + n(B) + n(C)
7. n(A’) = n(U) - n(A)
8. (a) n(A) = n[(A B’) (A B)] = n[(A - B) (A B)] = n(A - B) + n(A B)
(b) n(B) = n(B - A) + n(A B)
9. n(A B) = n(A - B) + n(A B) + n(B - A)
6.5 References
1) J K Sharma, ”Discrete Mathematics”
2) Kenneth H.Rosen, “Discrete Mathematics and its Applications”
3) Seymour Lipschutz and Marc Lipson, “Discrete Mathematics”
4) Charles D.Miller and Others, “Mathematical Ideas”
5) Wikipedia, the free encyclopedia.
Unit – III
Mathematical Logic
Contents
7.0 Aims and objectives
7.1 Introduction
7.2 Propositional calculus
7.2.1 Propositions
7.2.2 Types of Propositions
7.2.3 Basic Logical Operations
7.2.4 Derived Connectives
7.3 Conditional statement
7.3.1 Special characteristics of conditional statements
7.3.2 Converse, Inverse and Contrapositive
7.4 Biconditional statement
7.5 Problems and Solutions
7.6 Let Us Sum Up
7.7 Lesson End Activities
7.0 Aims and objectives
In this lesson, we have discussed the basis of propositional logic. With this logic,
students can found the truth and falsity of the statements.
After reading this lesson, you should able to understand
What is a propositional logic? and its usage in reasoning.
Types of statements
Connectives for forming compound statements
7.1 Introduction
Logic, logical thinking, and correct reasoning have wide applications in many
fields, including law, psychology, rhetoric, science, and mathematics. While an
interesting study can be made of logic in human lives, we shall restrict our attention
mainly to logic as it is used in mathematics. This logic was first studied systematically by
Aristotle (384 B.C.-322 B.C.). Aristotle and his followers studied patterns of correct and
incorrect reasoning.
Medieval philosophers and theologians, who made an intimate study of logical
arguments, carried the work of Aristotle forward. A big advance in the study of
mathematical logic came with the work of Gottfried Wilhelm von Leibniz (1646-1716),
one of the inventors of calculus. Leibnitz introduced symbols to represent ideas in logicletters
for statements and other symbols for the relations between statements. Leibnitz
hoped that logic would become a universal characteristic and unify all of mathematics.
Logic is the tool for reasoning about the truth and falsity of statements. There are
two main directions in which logic develops.
• The first is the depth to which we explore the structure of statements. The study
of the basic level of structure is called propositional logic. First order predicate logic,
which is often called just predicate logic, studies structure on a deeper level.
• The second direction is the nature of truth. For example, one may talk about
statements that are usually true or true at certain times.
“True” and “false” could be replaced by T and F (or any other two symbols) in our
discussions. Using T and F relates logic to Boolean functions. In fact, propositional logic
is the study of Boolean functions, where T plays the role of “true” and F the role of
“false.”
Our study is restricted to propositional and predicate logic only.
7.2 Propositional calculus
7.2.1 Propositions
A declarative sentence that is either true or false, but not both, is called a
proposition.
In mathematics, the propositions are denoted by alphabets known as propositional
variables. The conventionally used alphabets are p, q, r, s, and so on. The truth-value of a
proposition is true, denoted by T, if it is a true proposition and false, denoted by F, if it is
a false proposition. Here the letters T and F are constants.
Example for a true proposition:
Chennai is the state capital of Tamil Nadu. --- (1)
The truth-value of the above statement is true, dented by T.
Example for a false proposition:
Oxygen is a solid. --- (2)
The truth-value of the above statement is false, dented by F.
The area of logic that deals with propositions is called the propositional calculus
or propositional logic.
7.2.2 Types of Propositions
There are two types of propositions namely
1) Simple proposition and
2) Compound proposition.
A simple proposition is one in which the sentences cannot be further broken into simple
or atomic sentences. Two or more simple propositions connected by operators is known
as a compound proposition. The operators are known as logical connectives or simply
connectives. The logical connectives are shown in the table 3.1.
Table 3.1 - Logical connectives.
Symbol Connective Type of statement
not Negation
and Conjunction
or Disjunction
implies implication or conditional
if and only if equivalence or biconditional
Truth Table: It shows the relationship between the truth-value of a compound
proposition and the truth-values of its constituent simple propositions.
7.2.3 Basic Logical Operations
The basic logical operations are conjunction, disjunction and negation.
Conjunction
Let p and q be propositions. The proposition “p and q”, denoted by p q, is true
when both p and q are true and is false otherwise. The statement p q is called the
conjunction of p and q.
The truth table for the conjunctions of two propositions is given in the table 3.2.
Table 3.2
pqpÙq
TTT
TFF
FTF
FFF
Illustration
a) Consider the following four compound statements
1) Bharathiar University is at Coimbatore and 2+2 = 4.
2) Bharathiar University is at Coimbatore and 2+2 = 5.
3) Bharathiar University is at Chennai and 2+2 = 4.
4) Bharathiar University is at Chennai and 2+2 = 5.
Only the first statement is true. Each of the other statements is false, since at least
one of its simple statements is false.
Disjunction
Let p and q be propositions. The proposition “p or q”, denoted by p q, is false
when both p and q are false and is true otherwise. The statement p q is called the
disjunction of p and q.
The truth table for the disjunctions of two propositions is given in the table 3.3.
Table 3.3
pqpÚq
TTT
TFT
FTT
FFF
Illustration
b) Consider the following four compound statements
1) Bharathiar University is at Coimbatore or 2+2 = 4.
2) Bharathiar University is at Coimbatore or 2+2 = 5.
3) Bharathiar University is at Chennai or 2+2 = 4.
4) Bharathiar University is at Chennai or 2+2 = 5.
Only the last statement is false. Each of the other statements is true, since at least one of
its simple statements is true.
The disjunction may be either Inclusive or Exclusive. In the inclusive disjunction
the compound statement p q is true only when at least one of the statement is true. But
in the exclusive disjunction, commonly called inequivalence, the compound statement p
q is true only when either p or q but not both, is true.
Example 1
Let p : ABC Company earned 20% profit per share in 2005.
q : ABC Company paid 12% dividend per share in 2005.
The inclusive disjunction of p and q is
p q : ABC Company earned 20% profit per share in 2005 or ABC Company
paid 12% dividend per share in 2005 or both.
The exclusive disjunction of p and q is
p q : ABC Company earned 20% profit per share in 2005 or ABC Company
paid 12% dividend per share in 2005 but not both.
Negation
The negation of a true statement is false, and the negation of a false statement is
true. Its truth table is given in the table 3.4.
Table 3.4
p p
TF
FT
7.2.4 Derived Connectives
NAND
It is obtained by negating the result of ANDing of two statements.
For example, If p and q are two statements then NANDing of these two statements,
denoted by p q,
is false when both p and q are true, otherwise true. Its truth table is
given in the table 3.5.
Table 3.5
pqpq
TTF
TFT
FTT
FFT
NOR
It is obtained by negating the result of ORing of two statements.
For example, if p and q are two statements then ORing of these two statements, denoted
by p q, is true when both p and q are false, otherwise false. Its truth table is given in the
table 3.6.
Table 3.6
pqp¯q
TTF
TFF
FTF
FFT
XOR
If p and q are two statements then XORing of these two statements, denoted by p
q, is false when both p and q are same, otherwise true. Its truth table is given in the
table 3.7.
Table 3.7
pqpÅq
TTF
TFT
FTT
FFF
7.3 Conditional statement
A conditional statement is a compound statement that uses the connective
if…then. For example, the statement
If I read for too long, then I get a headache.
In the above statement, the component coming after the word if gives a condition (but not
necessarily the only condition) under which the statement coming after then will be true.
The conditional is written with an arrow, so that “if p, then q” is symbolized as
pq. We read p q as “p implies q” or “if p, then q”. In the conditional p q, the
statement p is the antecedent, while q is the consequent.
The conditional connective may not always be explicitly stated. That is, it may be
“hidden” in an everyday expression. For example, the statement
“It is difficult to study when you are distracted”
can be written
“It you are distracted then it is difficult to study”.
Truth table for the conditional “if p, then q” is given in 3.8
Table 3.8
p q p q
TTT
TFF
FTT
FFT
7.3.1 Special characteristics of conditional statements
1) p q is false only when the antecedent is true and the consequent is false.
2) If the antecedent is false, then p q is automatically true.
3) If the consequent is true, then p q is automatically true.
Negation of p ® q
The negation of p q is p q.
Conditional as a disjunction
A conditional may be written as a disjunction as below.
p ® q is equivalent to Ø p Ú q.
7.3.2 Converse, Inverse and Contrapositive
An conditional statement is made up of an antecedent and a consequent. If they
are interchanged, negated or both, a new conditional statement is formed. Suppose that
we begin with the direct statement
“If you stay, then I go,”
and interchange the antecedent (“you stay”) and the consequent (“I go”). We obtain the
new conditional statement
“If I go, then you stay.”
This new conditional is called the converse of the given statement.
By negating both the antecedent and the consequent, we obtain the inverse of the
given statement.
“If you do not stay, then I do not go.”
If the antecedent and the consequent are both interchanged and negated, the
contrapositive of the given statement is formed:
“If I do not go, then you do not stay.”
These three related statements for the conditional p q are summarized below.
Related conditional Statements
Direct statement p q (If p, then q.)
Converse q p (If q, then p.)
Inverse p q (If not p, then not q.)
Contrapositive q p (If not q, then not p.)
7.4 Biconditional statement
In elementary algebra we learn that both of these statements are true:
If x > 0, then 5x > 0.
If 5x > 0, then x > 0.
Notice that the second statement is the converse of the first. If we wish to make the
statement that each condition (x > 0, 5x > 0) implies the other, we use the following
language:
x > 0 if and only if 5x > 0.
This also may be stated as
5x > 0 if and only if x > 0.
The compound statement p if and only if q is called a biconditional. It is
symbolized p q, and is interpreted as the conjunction of the two conditionals p q
and q p. Using symbols, this conjunction is written
(p q) (q p )
so that, by definition,
p q (p q) (q p ).
Using this definition, the truth table for the biconditional p q can be determined as
shown in table 3.9.
Table 3.9
pqp«q
TTT
TFF
FTF
FFT
Example 2
Tell whether each biconditional statement is true or false.
(a) 6+9= 15 if and only if 12+4= 16
Both 6 + 9 = 15 and 12 + 4 = 16 are true. By the truth table for the biconditional,
this biconditional is true.
(b) 5 + 2 = 10 if and only if 17 + 19 = 36 .
Since the first component (5 + 2 = 10) is false, and the second is true, the entire
biconditional statement is false.
(c) 6=5 if and only if 1212
Both component statements are false, so by the last line of the truth table for the
biconditional, the entire statement is true. (Understanding this might take some extra
thought!) .
7.5 Problems and Solutions
1) Make truth tables for
(a) (p q) (p r) (b) p q
r
(c) p q r
Solution (a) Truth table is shown in Table 3.10
Truth table 3.10: (p q) (p r)
p Q r (p q) (p r) (p q) (p r)
TTTFFF
TTFFFF
TFTFFF
TFFFFF
FTTFFF
FTFFTF
FFTTFF
FFTTTT
(b) The truth table is shown in Table 3.11
Truth table: 3.11 p q
r
pQrpq
pq
r
TTTFT
TTFFT
TFTTF
TFFTT
FTTTF
FTFTT
FFTTF
FFFTT
(c) The truth table is shown in Table 3.12.
Truth table: 3.12 p Å q Å r
pqrpÅqpÅqÅr
TTTFT
TTFFF
TFTTF
TFFTT
FTTTF
FTFTT
FFTFT
FFFFF
2) If p and q are two statements, then show that the statement (p q
) (p q
) is
equivalent to (p q) (p q).
Solution: The equivalence of two compound statements is shown in the truth Table 3.13.
Truth table: 3.13 (p q)
(p q)
and (p q) (p q)
pQpq
(p q)
(p q)
(p q) (p q) (p q) (p q)
(1) (2) (3) (4) (5) (6) (7)
TTFFTFF
TFTFTFF
FTTFTFF
FFTFFTF
Since values in Columns (4) and (7) are same, therefore two statements are equivalent.
3) If p and q are two statements, then show that p q is equivalent to (p q)(p g).
Solution: The equivalence of two compound statements is shown in truth Table 3.14.
Truth table 3.14 : p q and (p q) (p g)
p q p q p q p q p g (p q) (p g)
(1) (2) (3) (4) (5) (6) (7) (8)
TTFFFFFF
TFTTTTFT
FTTFFFTT
FFFTTFFF
Since values in Columns (3) and (8) are same, therefore two compound statements are
equivalent.
4) If p and q are two statements, then show that the statement (p q) (p q) is
equivalent to pq.
Solution: The equivalence of two compound statements is shown in truth Table 3.15.
Truth table: 3.15 (p q) (p q) and pq
p q (p q) (p q) (p q) (p q) pq
(1) (2) (3) (4) (5) (6)
TTFFFF
TFTFTT
FTTFTT
FFFTTT
Since values in Columns (5) and (6) are same, therefore two compound statements are
equivalent.
7.6 Let Us Sum Up
Symbols Used in this Chapter
Connectives Symbols Types of
Statements
and Conjunction
or Disjunction
not Negation
if. . . then Conditional
if and only if Biconditional
Truth Tables
p p
TF
FT
pqpÙqpÚqpq
p ¯ q p Å q p q p « q
TTTTFFFTT
TFFTTFTFF
FTFTTFTTF
FFFFTTFTT
Statements Related to Conditional
Direct statement p q (If p, then q.)
Converse q p (If q, then p.)
Inverse p q (If not p, then not q.)
Contrapositive q p (If not q, then not p.)
7.7 Lesson End Activities
Write a negation for each of the following statements.
1. 5+3=9
2. Every good boy deserves favour.
3. Some people here can't play this game.
4. If it ever comes to that, I won't be here.
5. My mind is made up and you can't change it.
Let p represent "it is broken" and let q represent "you can fix it." Write each of the following
in symbols.
6. If it isn't broken, then you can fix it.
7. It is broken or you can't fix it.
8. You can't fix anything that is broken.
Using the same directions as for Exercises 6-8, write each of the following in words.
9. p q 10. p q
In each of the following, assume that p and q are true, with r false. Find the truth value of
each statement.
11. p r 12. r (pq)
13. r (sr) (The truth value of the statement s is unknown.)
14. r (pq)
15. What are the necessary conditions for a conditional statement to be false? for a conjunction
to be true?
16. Explain in your own words why, if p is a statement, the biconditional pp must
be false.
Write a truth table for each of the following. Identify any tautologies.
17. p (p q) 18. (pq)(pq)
Decide whether each statement is true or false.
19. All positive integers are whole numbers.
20. If x + 4 = 6, then x > 1.
Write each conditional statement in the form if. . . then.
21. All rational numbers are real numbers.
22. Being a rectangle is sufficient for a polygon to be a quadrilateral.
23. Being divisible by 2 is necessary for a number to be divisible by 6.
24. She cries only if she is hurt.
For each statement, write (a) the converse, (b) the inverse, and (c) the contrapositive.
25. If a picture paints a thousand words, the graph will help me understand it.
26. p(qr) (Use one of De Morgan's laws as necessary.)
Tautology, Contradiction and Arguments
Contents
8.0 Aims and Objectives
8.1 Introduction
8.2 Tautology
8.3 Contradiction
8.4 Arguments
8.4.1 Valid and Invalid Arguments
8.4.2 Using Truth Tables to Analyze Arguments
8.5 Problems and Solutions
8.6 Chapter Summary
8.7 Lesson End Activities
8.8 References
8.0 Aims and Objectives
The main aim and objective of this lesson is to know about the Tautology,
Contradiction and Arguments.
After reading this unit, students should be able to
Identify the statements that are Tautology or Contradiction.
Test the validity of this argument
8.1 Introduction
In this lesson we are going to discuss about the certain compound propositions
that are always true or false irrespective of the truth values of its component propositions.
The statement that is always true is called a tautology. The statement that is always false
is called Contradiction. We also discussed the method of testing the validity of an
argument.
8.2 Tautology
Definition: A statement that represents the constant T is called a tautology. In other
words, the statement is true for all truth values of the statement variables i.e. a tautology
contain only T in the last column of their truth table.
For example, the proposition “ p or not p”, that is, p p, is a tautology.
Table
p p p Ú Ø p
TFT
FTT
Example1: Show that ((p)(q)) p is a tautology
P q Ø p Ø q (Ø p)Ú (Ø q) ((Ø p)Ú (Ø q)) Ú p
TTFFFT
TFFTTT
FTTFTT
FFTTTT
8.3 Contradiction
Definition: A statement that represents the constant F is called a contradiction. In other
words, the statement is false for all truth values of the statement variables i.e. a
contradiction contain only F in the last column of their truth table.
.
Example 2: The proposition “ p and not p”, that is, p p, is a contradiction.
Table
p p p Ù Ø p
TFF
FTF
Note that the negation of a tautology is a contradiction since it is always false, and
the negation of a contradiction is a tautology since it is always true.
8.4 Arguments
There are two types of reasoning: inductive and deductive. So far we have concentrated
on using inductive reasoning to observe patterns and solve problems. Now, in this section
and the next, we will study how deductive reasoning may be used to determine whether
logical arguments are valid or invalid. A logical argument is made up of premises
(assumptions, laws, rules, wide ideas, or observations) and a conclusion. Together, the
premises and the conclusion make up the argument. Also recall that deductive reasoning
involves drawing specific conclusions from given general premises. When reasoning
from the premises of an argument to obtain a conclusion, we want the argument to be
valid.
8.4.1 Valid and Invalid Arguments
An argument is valid if the fact that all the premises are true forces the conclusion
to be true. An argument that is not valid is invalid, or a fallacy.
It is very important to note that "valid" and "true" are not the same - an argument can be
valid even though the conclusion is false (See Example below).
Definition
An argument is an assertion that a given set of propositions P1, P2, …, Pn called
premises, yields ( has a consequence) another proposition Q, called the conclusion. Such
an argument is denoted by
Several techniques, like Euler diagrams (visual technique) and the method of truth
table, can be used to check whether an argument is valid. In this section the method of
truth tables is shown.
8.4.2 Using Truth Tables to Analyze Arguments
While Euler diagrams often work well for simple arguments, difficulties can
develop with more complex ones. These difficulties occur because Euler diagrams
require a sketch showing every possible case. In complex arguments it is hard to be sure
that all cases have been considered.
In deciding whether to use Euler diagrams to test the validity of an argument look
for quantifiers such as "all," "some," or "no." These words often indicate arguments best
tested by Euler diagrams. If these words are absent, it may be better to use truth tables to
test the validity of an argument.
Q
P , P , , P1 2 n 
Example 3: Consider the following argument:
If the floor is dirty, then I must mop it.
The floor is dirty.
-----------------------------------------------
I must mop it.
In order to test the validity of this argument, we begin by identifying the simple
statements found in the argument. They are "the floor is dirty" and "I must mop it”. We
shall assign the letters p and q to represent these statements:
p represents "the floor is dirty."
q represents "I must mop it."
Now we write the two premises and the conclusion in symbols:
Premise 1: p q
Premise 2: p
---------------------Conclusion: q
To decide if this argument is valid, we must determine whether the conjunction of
both premises implies the conclusion for all possible cases of truth values for and q.
Therefore, write the conjunction of the premises as the antecedent of a conditional
statement, and the conclusion as the consequent.
[(p q) p] q
premise
and premise implies conclusion
p q p ® q (p ® q) Ù(p) [(p ® q) Ù p] ® q
TTTTT
TFFFT
FTTFT
FFTFT
Since the final column, indicates that the conditional statement that represents the
argument is true for all possible truth values of p and q, the statement is a tautology.
Thus, the argument is valid.
The pattern of the argument in the example 3 (floor-mopping),
p q
p
-------q
is a common one, and is called modus ponens, or the law of detachment.
In summary, to test the validity of an argument using truth tables, follow the steps in
the box that follows.
Testing the Validity of an Argument with a Truth Table
1. Assign a letter to represent each simple statement in the argument.
2. Express each premise and the conclusion symbolically.
3. Form the symbolic statement of the entire argument by writing the conjunction of all
the premises as the antecedent of a conditional statement, and the conclusion of the
argument as the consequent.
4. Complete the truth table for the conditional statement formed in part 3 above. If it is a
tautology, then the argument is valid; otherwise, it is invalid.
8.5 Problems and Solutions
1) Use the truth table to determine whether the statement
((p)q) (p (q)) is a tautology
P q Ø p Ø q (Ø p)Ú q p Ù(Ø q) ((Ø p)Ú q) Ú (p Ù(Ø q))
TTFFTFT
TFFTFTT
FTTFTFT
FFTTTFT
The last column contains only T. Therefore the given statement is a tautology.
2) Determine whether the argument is valid or invalid.
If my check arrives in time, I'll register for the Fall semester.
I've registered for the Fall semester.
-------------------------------------------------------------------------My check arrived in time.
Let p represent "my check arrives (arrived) in time" and let q represent "I'll register
(I've registered) for the Fall semester." Using these symbols, the argument can be written
in the form
p q
q
-------p
To test for validity, construct a truth table for the statement [(p q) q] p.
p q p ® q (p ® q) Ù(q) [(p ® q) Ù q] ® p
TTTTT
TFFFT
FTTTF
FFTFT
The third row of the final column of the truth table shows F, and this is enough to
conclude that the argument is invalid.
If a conditional and its converse were logically equivalent, then an argument of
the type found in the above example would be valid. Since a conditional and its converse
are not equivalent, the argument is an example of what is sometimes called the fallacy of
the converse.
3) Determine whether the argument is valid or invalid.
If a man could be two places at one time, I'd be with you.
I am not with you.
-------------------------------------------------------------------A man can't be two places at one time.
If p represents "a man could be two places at one time" and q represents "I'd be with
you," the argument becomes
p q
q
-------p
The symbolic statement of the entire argument is
[(p q) q] p
The truth table for this argument, shown below, indicates a tautology, and the argument is
valid.
p q p ® q Ø q (p ® q) Ù(Ø q) Ø p [(p ® q) Ù Ø q]® Ø p
TTTFFFT
TFFTFFT
FTTFFTT
FFTTTTT
The pattern of reasoning of this example is called modus tollens, or the law of
contraposition, or indirect reasoning.
With reasoning similar to that used to name the fallacy of the converse, the
fallacy
p q
p
-------q
is often called the fallacy of the inverse. An example of such a fallacy is "If it rains, I get
wet. It doesn't rain. Therefore, I don't get wet."
4) Determine whether the argument is valid or invalid.
I'll buy a car or I'll take a vacation.
I won't buy a car.
----------------------------------------I'll take a vacation.
If p represents "I'll buy a car" and q represents "I'll take a vacation," the argument
becomes
p q
p
-------q
We must set up a truth table for
[(p q) p] q
p Q p Ú q Ø p (p Ú q) Ù(Ø p) [(p Ú q) Ù Ø q]® q
TTTFFT
TFTFFT
FTTTTT
FFFTTT
The statement is a tautology and the argument is valid. Any argument of this form is
valid by the law of disjunctive syllogism.
5) Determine whether the argument is valid or invalid.
If it squeaks, then I use WD-40.
If I use WD- 40, then I must go to the hardware store.
---------------------------------------------------------------If it squeaks, then I must go to the hardware store.
Let p represent "it squeaks," let q represent "I use WD-40," and let r represent "I must go
to the hardware store." The argument takes on the general form
Make a truth table for the following statement:
p q
q r
-------p r
It will require eight rows.
p q r p® q q®r p ® r (p ® q) Ù(q®r) [(p ® q) Ù(q®r)] ® (p ® r)
TTTTTTTT
TTFTFFFT
TFTFTTFT
TFFFTFFT
FTTTTTTT
FTFTFTFT
FFTTTTTT
FFFTTTTT
This argument is valid since the final statement is a tautology. The pattern of argument
shown in this example is called reasoning by transitivity, or the law of hypothetical
syllogism. .
8.6 Chapter Summary
Tautology
A statement is true for all truth values of the statement variables is called a
tautology.
Contradiction
A statement is false for all truth values of the statement variables is called a
contradiction.
Argument
A logical argument is made up of premises (assumptions, laws, rules, wide ideas,
or observations) and a conclusion and denoted by.
Valid and Invalid Arguments
An argument is valid if the fact that all the premises are true forces the conclusion
to be true. An argument that is not valid is invalid, or a fallacy.
A summary of the valid and invalid forms of argument presented in this lesson follows.
Valid Argument Forms
Modus
Ponens
Modus
Tollens
Disjunctive
Syllogism
Reasoning by
Transitivity
p®q
p
------q
p®q
Øq
------Øp
pÚq
Øp
------q
p®q
q®r
-------
p®r
Invalid Argument Forms (Fallacies)
Fallacy of the
Converse
Fallacy of the
Inverse
p®q
q
------p
p®q
Øp
------Øq
Q
P , P , , P1 2 n 
8.7 Lesson End Activities
1. What is a hypothesis in an argument?
2. What is a premise in an argument?
3. What is a conclusion in an argument?
4. What is a valid argument?
5. What is an invalid argument?
6. State the modus ponens rule of inference.
7. State the modus tollens rule of inference.
8. State the addition rule of inference.
9. State the simplification rule of inference.
10. State the conjunction rule of inference.
Formulate the arguments of Exercises 11-15 symbolically and determine whether each is
valid. Let
p: I study hard. q: I get A's. r: I get rich.
11. If I study hard, then I get A's.
I study hard.
----------------------------------I get A's.
12. If I study hard, then I get A's.
If I don't get rich, then I don't get A's.
----------------------------------I get rich.
13. I study hard if and only if I get rich.
I get rich.
----------------------------------I study hard.
14. If I study hard or I get rich, then I get A's.
I get A's.
----------------------------------If I don't study hard, then I get rich.
15. If I study hard, then I get A's or I get rich.
I don't get A's and I don't get rich.
----------------------------------I don't study hard.
In Exercises 16-20, write the given argument in words and determine whether each
argument is valid. Let
p: 4 megabytes is better than no memory at all.
q: We will buy more memory,
r: We will buy a new computer.
16. p r
p q
---------------p ( rq)
17. p (rq)
r q
------------p r
18. p r
r q
---------q
19. r p
r
----------p
20. p r
r q
p
----------q
8.8 References
1) J K Sharma, ”Discrete Mathematics”
2) Kenneth H.Rosen, “Discrete Mathematics and its Applications”
3) Seymour Lipschutz and Marc Lipson, “Discrete Mathematics”
4) Charles D.Miller and Others, “Mathematical Ideas”
5) Wikipedia, the free encyclopedia.
Method of proof and Predicate Calculus
Contents
9.0 Aims and objectives
9.1 Introduction
9.2 Method of Proof
9.2.1 Direct Proof
9.2.2 Proof by contradiction (or Indirect Proof)
9.3 Predicate Calculus
9.4 Quantifiers
9.5 Mathematics and Predicate Calculus
9.6 Let Us Sum Up
9.7 References
9.0 Aims and objectives
In this lesson, we have discussed method of proofs and predicate calculus. With
the proof methods, one can arrive at a conclusion from the given sequence of premises.
After reading this lesson, you should able to understand
Direct and Indirect proof methods
Predicate calculus for reasoning.
Quantifiers and its types.
9.1 Introduction
An argument that establishes the truth of a theorem is called a proof. Logic is a
tool for the analysis of proofs. In this lesson, we describe some general methods of proof.
In the previous lesson we have discussed the propositional logic which is not sufficient
for all our logic needs. We need to go beyond the propositional calculus to the predicate
calculus, which allows us to manipulate statements and to express Mathematics. This and
other logics are employed in the design of expert systems, robots and artificial
intelligence. We have discussed predicate calculus in this lesson.
9.2 Methods of Proof
A mathematical system consists of axioms, definitions, and undefined terms.
Axioms are assumed true. Definitions are used to create new concepts in terms of
existing ones. Some terms are not explicitly defined but rather are implicitly defined by
the axioms. Within a mathematical system we can derive theorems. A theorem i s a
proposition that has been proved to be true. Special kinds of theorems are referred to as
lemmas and corollaries. An argument that establishes the truth of a theorem is called a
proof. Logic is a tool for the analysis of proofs. In this section, we describe some general
methods of proof.
Theorems are often of the form
For all x1,x2,…,xn, if p(x1,x2,…,xn), then q((x1,x2,…,xn).
This universally quantified statement is true provided that the conditional
proposition
if p(x1,x2,…,xn), then q((x1,x2,…,xn) ----- (7.1)
is true for all x1,x2,…,xn in the domain of discourse. To prove (7.1) we assume that
x1,x2,…,xn are arbitrary members of the domain of discourse. If p(x1,x2,…,xn), by
definition of p q, (7.1) is true; thus we need to consider the case that p(x1,x2,…,xn) is
true.
9.2.1 Direct Proof
A direct proof assumes that p(x1,x2,…,xn) is true and then, using p(x1,x2,…,xn) as
well as other axioms, definitions and theorems, shows directly that q(x1,x2,…,xn) is true.
Example 1: We will give a direct proof of the following statement. For all real numbers
d,d1,d2 and x
if d = min{d1,d2} and x <= d, then x <=d1 and x <=d2.
Proof.
We assume that d,d1,d2 and x are arbitrary real numbers. The preceding discussion
shows that it suffices to assume that
d = min{d1,d2} and x <= d
is true and then prove that
x <= d1 and x <=d2
is true.
From the definition of min, it follows that d <=d1 and d <=d2. From x <= d and
d<=d1, we may derive x <= d1 from the theorem – for all real numbers x,y, and z if x <= y
and y <= z, then x <= z. From x <= d and d <= d2, we may derive x <= d2 from the same
previous theorem. Therefore, x <=d1 and x <=d2.
Example 2 Let the following statements be true.
It is snowing.
If it is warm, then it is not snowing.
If it is not warm, then I cannot go for swimming.
Show that the statement 'I cannot go for swimming' is a true statement.
Solution Let p, q and r represent the statements.
p : it is snowing
q : it is warm
r : I can go for swimming
Consider premises that the statements
p, q p and q r
are true. Then we shall prove that the statement r is true.
Since q = p is true, its contrapositive p = r is also true. Since p q and q
r, then as per the law of syllogisms p r. Since statement p is true, therefore, the
statement - r: I cannot go for swimming, must be true.
Example 3 Let the following statements be true.
If I enjoy studying, then I will study.
I will do my homework or I will not study.
I will not do my homework.
Show that the statement 'I do not enjoy studying' is a true statement.
Solution Let p, q and r represent the statements.
p : I enjoy study
q : I will study
r : I will do my homework
Consider premises that the statements
p q, Ø q r and r
are true. Then we shall prove that the statement p is true.
Since r is true, then r is false. Also either r or q is true. Thus, q is true. Since p
q, therefore, qp is true and hence, the statement p: I do not enjoy study, must
be true.
9.2.2 Proof by contradiction (or Indirect Proof)
A second technique of proof is proof by contradiction. A proof by contradiction
establishes (7.1) by assuming that the hypothesis p is true and that the conclusion q is
false and then, using p and q’ as well as other axioms, definitions and theorems, devices a
contradiction. A contradiction is a proposition of the form r r (r may be any
proposition whatever). A proof by contradiction is sometimes called an indirect proof
since to establish (7.1) using proof by contradiction, one follows an indirect route: derive
r r, then conclude that (7.1) is true.
The only difference between the assumptions in a direct proof and a proof by
contradiction is the negated conclusion. In a direct proof the negated conclusion is not
assumed, where as in a proof by contraciction the negated conclusion is assumed.
Proof by contradiction may be justified by noting that the propositions p q and
p q r r are equivalent. The equivalence is immediate from a truth table.
p q r p® q p Ù Øq r Ù Ør p Ù Øq ® r Ù Ør
TTTTFFT
TTFTFFT
TFTFTFF
TFFFTFF
FTTTFFT
FTFTFFT
FFTTFFT
FFFTFFT
Example 4:
We will give a proof by contradiction of the following statement:
For all real numbers x and y, if x+y 2, then either x 1 or y 1.
Proof:
Suppose that the conclusion is false. Then x < 1 and y < 1.
Using a above mentioned theorem, we may add these inequalities to obtain
x+y < 1+1 =2.
At this point, we have derived the contradiction p p, where p: x+y 2.
Thus we conclude that the statement is true.
Example 5 Let the following statements be true.
. If I am lazy, then I do not study.
. I study or I enjoy myself.
. I do not enjoy myself.
Show that the statement 'I am not lazy' is a true statement.
Solution Let p, q and r represent the statements
p : I am lazy
q : I study
r : I enjoy myself
Consider premises that the statements
p q, q r and r
are true. Then we shall prove that the statement p is true.
Assuming that p is true. Since p is true and p q, then q is true. Thus, q is false.
Also, r is false. But either q or r is true. This is not possible This contradiction implies
that the assumption that p is true is false and hence p is true.
Example 6 Show that 5 + 2 is an irrational number.
Solution Let 5 + 2 = r be a rational number. Then 2 = r - 5.
Since r and 5 are rational numbers, therefore, r - 5 is also a rational number. This
implies that 2 is a rational number, which is a contradiction because 2 is an irrational
number. Thus, the assumption that 5 + 2 must be false and hence, the given number is
irrational.
9.3 Predicate Calculus
We begin our discussion with examples.
Example 7:
We have been studying statements that are either true or false. But, consider the
statement “x2 > 1.” In order to decide if this statement is true or false, we need to know
the numerical value of x. If x = 1.1, then “x2 > 1” is true. If x = 0.9, then “x2 > 1” is false.
The best way to think of this is to regard the statement “x2 > 1” as a function S(x) =
“x2>1”. If we take this point of view, we need to specify the domain of S. First suppose
the domain of S is R, the set of all real numbers. The codomain (or range) of S, by our
description just given, is a set of statements that are either true or false (e.g., S(0.9) =
“0.92 > 1”, S(2.3) = “2.32 > 1”). The function S is an example of a predicate.
Example 8:
Consider the statement "All men are mortal" and reword the statement like
"For all x, if x is a man then x is mortal."
The sentence "x is a man" is not a statement in propositional calculus, since it involve an
unknown thing x and we can't assign a truth value without knowing what x we're talking
about. This sentence can be broken down into its subject, x, and a predicate, "is a man."
We say that the sentence is a statement form, since it becomes a statement once we fill
in x. Here is how we shall write it symbolically: The subject is already represented by the
symbol x, called a term here, and we use the symbol P for the predicate "is a man." We
then write Px for the statement form. (It is
traditional to write the predicate before the term; this is related to the convention of
writing function names before variables in other parts of mathematics.) Similarly, if we
use Q to represent the predicate "is mortal" then
Qx stands for "x is mortal." We can then
write the statement "If x is a man then x is mortal" as Px 
Qx. To write our whole statement, "For all x,
if x is a man then x is mortal" symbolically, we need symbols for "For all x." We use the
symbol "" to stand for the words "for all"
or "for every." Thus, we can write our complete statement as
x[Px 
Qx].
The symbol "" is called a quantifier
because it describes the number of things we are talking about: all of them.
Specifically, it is the universal quantifier
because it makes a claim that something happens universally.
The square brackets around “ Px 
Qx” define what is called the scope of the
quantifier x. That is, they surround what it is
we are claiming is true for all x.
Definition (Predicate and truth set) A predicate is any function whose codomain is
statements that are either true or false. There are two things to be careful about:
• The codomain is statements not the truth value of the statements.
• The domain is arbitrary — different predicates can have different domains.
The truth set of a predicate S with domain D is the set of those x D for which S(x) is
true. It is written
{x D | S(x) is true} or simply {x | S(x)}.
In simple terms, the Predicate can be defined as
In general, a statement involving the n variables x1,x2,…,xn can be denoted by
P(x1,x2,…,xn).
A statement of the form P(x1,x2,…,xn) is the value of the propositional function P at
the n-tuble (x1,x2,…,xn), and P is called a predicate.
9.4 Quantifiers
The words and phrases like for all, each and no(ne) are called universal
quantifiers, while words and phrases like for some, there exists and (for) at least one are
called existential quantifiers. Quantifiers are used to indicate how many cases of a
particular situation exist.
The phrase “for all” the universal quantifier is written (“A” rotated 180 ). If
S(x) is a predicate and the set D is contained in the domain of x, the statement “x D,
S(x)” is read “for all x D, S(x) is true,” or just “for all x D, S(x).” The statement “x
D, S(x)” is true if and only if S(x) is true for every x D; otherwise the statement “x
D, S(x)” is false. If the value of D is clear, we may write simply x S(x).
The phrase “for some” is called the existential quantifier and is written (“E”
rotated 180 ). If S(x) is a predicate and the set D is contained in the domain of x, the
statement “ x D, S(x)” is read “for some x D, S(x) is true,” or just “for some x D,
S(x).” It is also read “there exists x D such that S(x).” The statement “ x D, S(x)” is
true if and only if S(x) is true for at least one x D; otherwise the statement “ x D,
S(x)” is false. If the value of D is clear, we may write simply x S(x).
In terms of truth sets:
• “x D, S(x)” is equivalent to saying that the truth set of S(x) contains the set D.
• “x D, S(x)” is equivalent to saying that the truth set of S(x) contains at least one
element of the set D
Negation
Let us go back for a moment to the statement "All men are mortal":
x[Px 
Qx].
Question
What does its negation, x[Px

Qx]
, say?
Answer
Literally, it says: "It is not true that all men are mortal." More succinctly,
"Some men are immortal."
Contrast this with "No men are mortal":
x[Px 
Qx].
9.5 Mathematics and Predicate Calculus
Mathematics is expressed in the language of the predicate calculus.
Here's an example of a mathematical
statement expressed symbolically.
Example 9: A Mathematical Statement
Write the following statement symbolically:
"If a number is greater than 1 then it is greater than 0."
Solution
Since this is a statement meant to be true of every number, we need to rephrase it
to make the universal quantifier obvious:
"For all x, if x is a number and x is greater than 1, then x is greater than 0."
Let us write N for the predicate "is a number" and use the standard notation ">"
for "is greater than." Our statement is then:
x[(Nx
(x >1)) 
(x >0)].
Notice that we put the phrases "x>1" and "x>0" in parentheses to make the meaning
clearer.
9.6 Let Us Sum Up
Axioms are assumed true.
A theorem is a proposition that has been proved to be true.
Special kinds of theorems are referred to as lemmas and corollaries.
An argument that establishes the truth of a theorem is called a proof.
Direct Proof
A direct proof of (p q) has the following form. First assume that p is true. Then
from this assumption draw one conclusion after another. Finally conclude that q is
true. Therefore (p q) must always be true.
Indirect Proof
An Indirect proof of (p q) has the following form. We assume that p is true,
and that q is false. We then draw one conclusion after another, until we arrive at some
statement r which we know is false. Contradiction, we say. So it cannot happen that p
is true and q is false. This means that whenever p is true, q must also be true.
Therefore (p q).
Predicate
A statement of the form P(x1,x2,…,xn) is the value of the propositional function P
at the n-tuble (x1,x2,…,xn), and P is called a predicate.
Predicate calculus
The area of logic that deals with predicates and quantifiers is called the predicate
calculus.
Quantifiers
The phrase “for all” the universal quantifier is written .
The phrase “for some” is called the existential quantifier and is written .
9.7 References
1) J K Sharma, ”Discrete Mathematics”
2) Kenneth H.Rosen, “Discrete Mathematics and its Applications”
3) Seymour Lipschutz and Marc Lipson, “Discrete Mathematics”
4) Charles D.Miller and Others, “Mathematical Ideas”
5) Wikipedia, the free encyclopedia.
6) Richard JohnsonBaugh, ”Discrete Mathematics”
Unit – IV
Relations
Contents
10.0 Aims and Objectives
10.1 Introduction
10.2 Product of sets
10.3 Binary Relations
10.4 Types of relations
10.5 Problems and Solutions
10.6 Chapter Summary
10.7 References
10.0 Aims and Objectives
In this lesson, we have discussed about the Product of sets, Binary Relations and
Types of relations.
After completing this lesson, you should able to
Find product of sets.
Know about the Binary Relations.
Know the different types of relations.
10.1 Introduction
In mathematics and computer science, there are many relations used to establish a
relation between pair of objects taken in a definite order. For example, “less than”, “is
parallel to”, “is a subset of”, and so on. A relation between two sets can be defined by
listing their elements as ordered pairs. In this lesson, we will discuss relations defined on
sets and the ways of representing finite relations along with their properties.
Relations will be defined by ordered pairs of elements. An ordered pair consists of
two elements a and b is denoted as (a, b). Here the element a is designated as the first
element and b as the second element. In particular
(a, b) = (c, d)
if and only if a = c and b = d. Thus, (a, b) (b, a). This contrasts with sets, where the
order of elements is irrelevant; for example, {3,5} = {5,3}.



n
i1
A A ... A or A
10.2 Product of sets
Consider two non-empty finite sets A and B. The set of all ordered pairs (a, b)
where a A and bB is called the product or Cartesian product of A and B. This
product is denoted by A B (read as “A cross B”). By definition
A B ={(a, b) : a A and bB}
Example 1
If A = {1,2} and B = {a, b, c}, then
A B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
B A = {(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)}
A2 = A A = {(1, 1), (1, 2), (2, 1), (2, 2)} and
12ni
n(A B) = n(A) . n(B) =2 . 3 = 6
From the above, we conclude the following:
i)A B B A i.e. the order in which the sets are considered is important.
ii) In fact, n(A B) = n(A) . n(B) for any finite sets A and B. This follows from
the observation that, for an ordered pair (a, b) in A B, there are n(A)
possibilities for a, and for each of these there are n(B) possibilities for b.
Product of n sets
The idea of a product of sets can be extended to any finite number of sets. Let A1,
A2, … , An be n given sets. The set of all ordered n - tubles (a1, a2, … ,an) where a1A1,
a2 A2, …, an An is called the product of the sets A1, A2, … , An and is denoted by
Important results
1) Let A, B, and C are sets and A B. Then (A C) = (B C).
Proof. Let (a, c) be an arbitrary element of A C. Then
(a, c)(A C) a A and c C
a B and c C
(a, c) B C
Thus, (A C) = (B C).
2) If A B and C D, then prove that (A C) (B D).
Proof. Let (a, c) be an arbitrary element of A C. Then
(a, c)(A C) a A and c C
a B and c D
(a, c) B D
Thus, (A C) (B C).
10.3 Binary Relations
Definition: Let A and B be sets. A binary relation or, simply, relation R from A to
B is a subset of A B.
In other words, a binary relation from A to B is a set R of ordered pairs where the
first element of each ordered pair comes from A and the second element comes from B.
Symbolically,
R : A B if and only if R A B and (a, b) R where a A and b B.
We use the notation a R b to denote that (a, b) R and a R b to denote that (a, b) R.
Moreover, when (a, b) belongs to R, a is said to related to b by R.
Number of relations between two sets
The total number of relations from A to B are 2mn, where m and n are the Cardinal
number of the sets A and B respectively, including the empty relation and the relation A
B itself.
Domain and range of a relation
The domain of a relation R is the set of all first elements of the ordered pairs,
which belong to R, and the range of R is the set of second elements.
Illustrations
a) Let A = {1, 2, 3} and B = {x, y, z} and let R = {(1,y), (1,z), (3,y)}. Then R is a
relation from A to B, since R is a subset of A B. With respect to this relation
1 R y, 1 R z, 3 R y but 1 R x, 2 R x, 2 R y, 2 R z, 3 R x, 3 R z.
The domain of R is {1,3} and its range is {y,2}
b) Let A = {eggs, milk, corn} and B = {cows, goats, hens}.
We can define a relation R from A to B by (a, b) R if a is produced by b. In
other words
R: {(eggs, hens), (milk, cows), (milk, goats)}
with respect to this relation
eggs R hens, milk R cows, etc.
Set operations on relations
Since all relations are set of ordered pairs, the set operations can be carried on
such sets. Let R and S be two relations, then we can define the following relation:
Intersection of R and S: x (R S) y = x R y
10.4 Types of relations
Inverse relation
Let R be a relation from a set A to set B. The inverse relation from B to A,
denoted by R-1, is the set of ordered pair {(b, a) : a A, b B, (a, b) R}.
For example, the inverse of the relation R={(1, y), (1, z), (3, y)} from A={1, 2, 3} to
B={x, y, z} follows:
R-1 = {(y,1), (z, 1), (y,3)}
From the above, if R is any relation then (R-1)-1= R. Also, the domain of R-1 is equal to
the range of R; the range of R-1 is equal to domain of R. Moreover, if R is a relation on A,
then R-1 is also a relation on A.
Identity (or Diagonal) Relation
Let A be any set. Then the relation R in a set A denoted by IA is said to be identity
relation if
IA={(a, a) : a A}.
Universal Relation
Let A be any set. Then A A which is a subset of A A is a relation on A called
the universal relation
For example, if A ={2,3} then R = A A = {(2,2), (2,3), (3,2), (3,3)} is universal
relation.
Empty (or Void) relation
A relation R in a set is said to be an empty relation provided R is a null set i.e.
R=.
10.5 Problems and Solutions
1) Consider the table given below that shows which students are taking which
courses.
Student Course
Bala CompSci
Manohar Math
Chandra Art
Ravi History
Ragu CompSci
David Math
If we let X ={Bala, Manohar, Ragu, David} and
Y ={CompSci, Math, Art, History,
Our relation R in table can be written
R={(Bala, CompSci), (Manohar, Math), (Chandra, Art),
(Ravi, History), (Ragu, CompSci), (David, Math)}.
Since (Ravi, History) R, we may write Ravi R History. The domain (First
Column) of R is the set X and the range (Second Column) of R is the set Y.
2) Let X={2,3,4} and Y={3,4,5,6,7}.
If we define a relation R from X to Y by (x,y) R if x divides y (with zero
remainder).
We obtain
R={(2,4),(2,6),(3,3),(3,6),(4,4)}.
If we rewrite R as table, we obtain
XY
24
26
33
36
44
The domain of R is the set {2,3,4} and the range of R is the set {3,4,6}.
3) Let R be the relation on X={1,2,3,4} defined by (x,y) R if x y., x,y X. Then
R={{1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,5)}
The domain and range of R are both equal to X.



n
i1
A A ... A or A
10.6 Chapter Summary
Product of sets
Let A and B be two non-empty finite sets. The set of all ordered pairs (a, b) where
aA and bB is called the product or Cartesian product of A and B. This product is
denoted by AB (read as “A cross B”). By definition
A B ={(a, b) : a A and bB}
Product of n sets
Let A1, A2, … , An be n given sets. The set of all ordered n - tubles (a1, a2, … ,an)
where a1A1, a2 A2, …, an An is called the product of the sets A1, A2, … , An
and is denoted by
Binary Relations
A binary relation or, simply, relation R from set A to set B is a subset of A B.
Number of relations between two sets
The total number of relations from A to B are 2mn, where m and n are the Cardinal
number of the sets A and B respectively, including the empty relation and the
relation A B itself.
Domain and range of a relation
The domain of a relation R is the set of all first elements of the ordered
pairs, which belong to R, and the range of R is the set of second elements.
Types of relations
Inverse relation
Let R be a relation from a set A to set B. The inverse relation from B to A,
denoted by R-1, is the set of ordered pair {(b, a) : a A, b B, (a, b) R}.
Identity (or Diagonal) Relation
Let A be any set. Then the relation R in a set A denoted by IA is said to be
12ni
identity relation if IA={(a, a) : a A}.
Universal Relation
Let A be any set. Then A A which is a subset of A A is a relation on A
called the universal relation.
Empty (or Void) relation
A relation R in a set is said to be an empty relation provided R is a null set
i.e. R=.
10.7 References
1. J K Sharma, ”Discrete Mathematics”
2. Kenneth H.Rosen, “Discrete Mathematics and its Applications”
3. Seymour Lipschutz and Marc Lipson, “Discrete Mathematics”
4. Charles D.Miller and Others, “Mathematical Ideas”
5. Wikipedia, the free encyclopedia.
6. Richard JohnsonBaugh, ”Discrete Mathematics”
Properties of relation, Partial Order and Equivalence relation
Contents
11.0 Aims and Objectives
11.1 Introduction
11.2 Properties of relations
11.3 Partial order relations
11.4 Equivalence relation
11.5 Composition of relations
11.6 Chapter Summary
11.7 References
11.0 Aims and Objectives
In this lesson, we have discussed about the Properties of relations, Partial order
relations, Equivalence relation, and Composition of relations.
After completing this lesson, you should able to
Know about Properties of relations.
Work with Partial order relations.
Compute Equivalence relation and Composition of relations.
11.1 Introduction
In this lesson we covered some important properties of the relations like Reflexive
relation, Symmetric relation, Anti-symmetric relation and Transitive relation. Using the
above properties we can define an equivalence relation which is also discussed in this
lesson.
11.2 Properties of relations
This section discusses some important properties or types of relations that are
defined on a given set A.
Reflexive relation
A relation R on a set A is called reflexive if (a, a) R for every a A. In other
words, for every a A, (a, a) R.
Illustrations
Consider the following relations on set A={1, 2, 3, 4}.
1) R1 = {(1, 1), (1, 2), (2, 3), (1, 3), (4, 4)}
2) R2 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}
3) R3 = A A , the universal relation
4) R4=, the empty relation.
Determine which of the relations are reflexive.
Here the set A contain the four elements 1, 2, 3 and 4 then a relation on A is reflexive if it
contains the four pairs (1,1), (2,2), (3,3) and (4,4). Therefore, the relation R1 is not
reflexive since 2 A but (2, 2) R1 and also R4 is not reflexive. Where as the relations
R2 and the universal relation R3 are reflexive.
Symmetric relation
A relation R on a set is symmetric if whenever aRb then bRa, that is, if whenever
(a, b)R, then (b, a) R. The necessary and sufficient condition for a relation R to be
symmetric is R = R-1.
Illustrations
1) Let A={1, 2, 3, 4} and R={(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)} then
R-1={(1, 1), (2, 1), (1, 2), (2, 2), (3, 3), (4, 4)}.
Here R = R-1. Therefore R is symmetric.
2) Let A ={1, 2, 3} and R={(1, 2), (2, 3), (2, 1), (3, 3)} then
R-1={(2, 1), (3, 2), (1, 2), (3, 3)}.
Here R R-1 since (2,3) R where as (3,2) R-1. Hence R is not symmetric.
Anti-symmetric relation
A relation R on a set A such that (a, b) R and (b, a) R only if a = b, for all
a,bR, is called anti-symmetric.
Transitive relation
A relation R on a set A is transitive if whenever (a, b) R and (b, c) R, then
(a,c) R, for all a,b,c A.
11.3 Partial order relations
A relation, R on a set A is called a partial order or partial ordering relation if and only if
following three conditions are satisfied
1) R is reflexive, i.e. a R a, for all a A.
2) R is anti-symmetric, i.e. a R b, b R a, if and only if a = b, for all a, b A.
3) R is transitive, i.e. a R b, b R c implies that a R c, for all a, b, c A.
The set A on which a partial order relation, R is defined is called a partially ordered
set or simply a poset, and it is denoted by (A, R).
The relation '=' (equal to) defined on a set A is said to be total order relation provided
for all a, b A, we have a = b. This implies that b = a. In other words, aRb and bRa, and
hence the relation '=' is also a symmetric relation.
11.4 Equivalence relation
An equivalence relation is a binary relation between two elements of a set which groups
them together as being "equivalent" in some way. That a is equivalent to b is denoted as
"a~b" or "a b". An equivalence relation is reflexive, symmetric, and transitive. In other
words, for all elements a, b, and c of the set X, the following must hold for "~" to be an
equivalence relation on X:
Reflexive: a ~ a
Symmetry: if a ~ b then b ~ a
Transitivity: if a ~ b and b ~ c then a ~ c.
A set X together with an equivalence relation on X is called a setoid.
Examples of equivalence relations
A ubiquitous equivalence relation is the equality ("=") relation between elements of any
set. Other examples include:
"Has the same birthday as" on the set of all people, given navie set theory.
"Is similar to" or "congruent to" on the set of all triangles.
"Is congruent to modulo n" on the integers.
"Has the same image under a function" on the elements of the domain of the
function.
Logical equivalence of statements in logic.
"Is isomorphic to" on models of a set of sentences.
Examples of relations that are not equivalences
The relation " " between real numbers is reflexive and transitive, but not
symmetric. For example, 7 5 does not imply that 5 7. It is, however, a partial
order.
The relation "has a common factor greater than 1 with" between natural numbers
greater than 1, is reflexive and symmetric, but not transitive. (The natural
numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a
common factor greater than 1, but 2 and 3 do not have a common factor greater
than 1).
The empty relation R on a non-empty set X (i.e. aRb is never true) is vacuously
symmetric and transitive, but not reflexive. (If X is also empty then R is
reflexive.)
The relation "is approximately equal to" between real numbers or other things,
even if more precisely defined, is not an equivalence relation, because although
reflexive and symmetric, it is not transitive, since multiple small changes can
accumulate to become a big change.
The relation "is a sibling of" on the set of all human beings is not an equivalence
relation. Although siblinghood is symmetric (if A is a sibling of B, then B is a
sibling of A) it is neither reflexive (no one is a sibling of himself), nor transitive
(since if A is a sibling of B, then B is a sibling of A, but A is not a sibling of A).
Instead of being transitive, siblinghood is "almost transitive", meaning that if A ~
B, and B ~ C, and A C, then A ~ C.
Symmetry and transitivity do not imply reflexivity, unless every A is related to
some B.
Theorem ("Fundamental Theorem of Equivalence Relations")
An equivalence relation ~ partitions X.
Conversely, corresponding to any partitions of X, there exists an equivalence
relation ~ on X.
11.5 Composition of relations
The composition of binary relations is a concept of forming a new relation S o R
from two given relations R a n d S, having as its most well-known special case the
composition of functions.
Properties
Composition of relations is associative.
The inverse relation of S o R is (S o R)-1 = R-1 o S-1.
The compose of (partial) functions (i.e. functional relations) is again a (partial)
function.
If R and S are injective, then S o R is injective, which conversely implies only the
injectivity of R.
If R and S are surjective, then S o R is surjective, which conversely implies only
the surjectivity of S.
The binary relations on a set X (i.e. relations from X to X) form a monoid for
composition, with the identity map on X as neutral element.
11.6 Let Us Sum Up
Properties of relations
Reflexive relation
A relation R on a set A is called reflexive if (a,a) R for every a A.
Symmetric relation
A relation R on a set A is called symmetric if for all a,b A, if (a,b) R,
then (b,a)R.
Antisymmetric relation
A relation R on a set A is called antisymmetric if for all a,b A, if (a,b) 
R and a b, then (b,a) R.
Transitive relation
A relation R on a set A is called transitive if for all a,b,z A, if (a,b) and
(y,z) R, then (x,z) R.
Partial order relations
A relation, R on a set A is called a partial order or partial ordering relation
if and only if following three conditions are satisfied
1) R is reflexive, i.e. a R a, for all a A.
2) R is anti-symmetric, i.e. a R b, b R a, if and only if a = b, for all a, b A.
3) R is transitive, i.e. a R b, b R c implies that a R c, for all a, b, c A.
Equivalence relation
A relation that is reflexive, symmetric and transitive on set A is called an
equivalence relation on A.
Composition of relations
Let R1 be a relation from X to Y, and R2 be a relation from Y to Z. The
composition of R1 and R2, denoted R2 o R1, is the relation from X to Z defined by
R2 o R1={(x,z) : (x,y) R1 and (y,z) R2 for some y Y}
11.7 References
1. J K Sharma, ”Discrete Mathematics”
2. Kenneth H.Rosen, “Discrete Mathematics and its Applications”
3. Seymour Lipschutz and Marc Lipson, “Discrete Mathematics”
4. Charles D.Miller and Others, “Mathematical Ideas”
5. Wikipedia, the free encyclopedia.
6. Richard JohnsonBaugh, ”Discrete Mathematics”
Functions
Contents
12.0 Aims and Objectives
12.1 Introduction
12.2 Functions as Relations
12.3 Difference Between a Function and its Value
12.4 Types of Functions
12.5 Composition of Functions
12.6 Lesson End Activities
12.7 References
12.0 Aims and Objectives
In this lesson, we have discussed about the Properties of relations, Partial order
relations, Equivalence relation, and Composition of relations.
After completing this lesson, you should able to
Know about Properties of relations.
Work with Partial order relations.
Compute Equivalence relation and Composition of relations .
12.1 Introduction
Many everyday phenomena involve two quantities that are related to each other
by some rule of correspondence. Such a rule of correspondence is called a function. The
concept of a function is extremely important in discrete mathematics. Functions are used
in the definition of such discrete structures as sequences and strings. Functions are also
used to represent how long it takes a computer to solve problems of a given size.
Recursive functions, which are functions defined in terms of themselves, are used
throughout computer science. This section reviews the basic concepts involving functions
needed in discrete mathematics. The terms such as mapping, transformation, etc., are also
used for functions to depict a relation between two discrete objects.
Definition of a Function
A function f from a set A to a set B is a rule of correspondence that assigns to
each element x in the set A exactly one element y in the set B. The set A is the
domain(or set of inputs) of the function f, and the set B contains the range(or set of
outputs). Symbolically, the function f is expressed as:
f : A B such that y = f(x), for x A and y B.
The following characteristics are true of a function from a set A to a set B.
1. Each element in A must be matched with an element in B.
2. Some elements in B may not be matched with any element in A.
3. Two or more elements of A may be matched with the same element of B.
The converse of the third statement is not true. That is, an element of A (the domain)
cannot be matched with two different elements of B.
In other words, the relation f can be described as the set of elements written as:
{(x, f(x)) : x A, f(x) B}.
Example 1: Testing for functions
Let A={a,b,c} and B={1,2,3,4,5}. Does the set of ordered pairs represent a function from
set A to set B?
i) {(a,2), (b,3), (c,4)} ii) {(a,4), (b,5)}
iii) {(a,1), (b,1), (c,1)} iv) { (a,1), (a,2), (b,3), (b,4), (c,5)}
i) Yes, because each element of A is matched with exactly one element of B.
ii) No, because not all elements of A are matched with an element of B.
iii) Yes, It does not matter that each element of A is matched with the same element
of B.
iv) No, because the element a in A is matched with two elements, 1 and 2, in B. This
is also true of the element b.
Illustrations
1) Let A be the set of books in a library and N be the set of natural numbers. Let f be the
rule which assigns to each book the number of pages contained in it. Here some
natural number will be assigned to each book and no book can be assigned to more
than one natural number. Therefore f is a function of A to N.
2) Let A = {l, 2, 3, 4} and B = {1, 2, 3}. Then f = {(1,2), (2, 3), (3, 3), (4, 2)} is a
function from A to B. Since to each element in A we have assigned a unique element
in B, therefore f(1) = 2, f(2) = 3, f(3) = 3, f(4) = 2. Also the range of the function is
f(A) = {2, 3}.
3) Let A = {1, 2, 3} and B = {x, y, z}. Consider the relations
R1 = {(1, x), (2, x)} and R2 = {(1, x), (1, y), (2, z), (3, y)}
The relation R2 is not a function because R2(1) = {x, y}. But relation R1 is a function
with Domain = {l, 2} and Range = {x}.
4. Let I be the set of integers and A = {0, 1}. The relation between I and A defined as
f: IA such that
is a function because each set f(x) consists of a single element.
12.2 Functions as Relations
A relation can be used to express a one-to-many relationship between the
elements of the sets A and B, where an element of A may be related to more than one
element of B. A function represents a relation where exactly one element of B is related
to each element in A. Relations are a generalization of functions; they can be used to
express a much wider class of relationships between sets. Thus, every function is a
relation but every relation is not a function.
Illustration Let A={a, b, c, d}, B = {1, 2, 3} and R = {(a, 1), (b, 1), (c, 2), (d, 2)}. Then
R is a function from A to B. Clearly, R is also a relation from A to B. But, consider the
subset C of A x B given by
C = {(a, 1), (b, 2), (a; 3), (c, 1), (d, 2)}.
Here C is a relation from A to B. But C is not a function from A to B. The obvious reason
is that the element a A is associated to two different elements 1, 3 B.
12.3 Difference between a Function and its Value
If f is the set of all ordered pairs of elements of A and B such that image of first element
is the second element, i.e. (x, y) f, then y is called the value of f at the element x and is
denoted by f(x). Hence, f is an element of A x B and f (x) is an element of B. For
example,
f = {x, y : y = 2x2 + 4x + 5}
defines the function. The numbers x and y are called variables. The domain of the
function is the set of all possible values of x and the range is the set of all values of y. In
this case the domain is the set of all real numbers which can be denoted with interval
notation as (, -), but the range of f is set of all positive numbers greater than or equal
to 3, i.e. (3, ) because lowest value that y can assume is 3 for x = -1.
Illustration If A = {1, 2, 3} and B = {2, 3, 4}, then the relation f defined by f(1) = 2, f(2)
= 3, f(3)= 4 is a function from set A to B. We may also write f = {(1, 2), (2,3), (3, 4)}.
Example If A = {I, 2, 3} and B = {2, 3, 4} are two sets, then show that {(I, 2), (1, 3), (3, 4)} is
not a function from A to B.
Solution The relation, f = {(1, 2), (1, 3), (3, 4)} implies that f(1) = 2, f(1) = 3 and f(3) = 4.


1 if x is odd
0 if x is even
f(x)
Clearly f is not a function for two reasons:
(i) The element 2 of A has no image in B.
(ii) The element 1 of A has two images, viz., 2 and 3 in B.
Identity function
Let A be any set. The function from A into A i.e. f : A A, which assigns to each
element that element itself is called the identity function on A and is usually denoted by IA
or simply I. In other words,
IA(a) = a
for every element a in A.
12.4 Types of Functions
Equal functions Two functions f : A B and g : A B are said to be equal if and only
if
f (x) = g (x) for every x A and are written as f = g.
Constant function A function f : AB is called a constant function, if some element
yB, is assigned to every element of A, i.e. f (x) = y for every x A. In other words,
f:AB is a constant function if the range of f consists of only one element.
Onto function The function f: A B is said to be an onto (or surjective) function, if
each element of B is the image of some element of A. In other words, f: A B is onto if
the image of f is the entire codomain, i.e., if f(A)=B. In such a case we say that f is
function from A onto B or that f maps A onto B.
Into function The function f : A B is said to be an into function, if there is at least one
element in B which is not the f-image of any element in A. But every element of A has fimage
in B. In this case the range of f is proper subset of B, that is, f (A) B and f (A)#B.
Many-one and one-to-one functions The function f: A B is said to be a
(a) Many-one function, if two or more different elements in A have same f-image in B,
i.e, f(x1)=f(x2) even if x1# x2 for all x1,x2 A.
(b) One-to-one (or injective) function, if different elements in the domain A have distinct
images. In other words, f is one-to-one if f(a) = f(a’) implies a=a’.
One-one onto (bijective) function The function f: AB is said to be one-one onto
function if to each element of A there corresponds one and only one element of B and
every element of B have one and only image in A. One-one and onto function are also
called bijective as shown in Fig. 12.1.
One-one into function The function f : A B is said to be one-one into function if to
each element of A there corresponds one element of B, but there are some element of B
which do not correspond to any of the elements of A as shown in Fig. 12.2.
Many-one into function The function, f : A B is said to be many-one into function if
two or more elements of a set A corresponds to the same element of B and there are some
elements in B which do not correspond to any of the elements of A as shown in Fig. 12.3.
Many-one onto function The function, f : A B is said to be many-one onto function if
each element in B is joined to at least one element in A, and two or more elements in A
are joined to the same element in B as shown in Fig. 12.4.
Illustrations Let A be the set of students sitting on chairs in a classroom and let B be the
set of chairs in the classroom. Let f be the correspondence, which associates to each
student the chair on which he sits.
Since every student has some chair to sit and no student can sit on two or more than
two chairs, therefore for the function defined as f: A B, the following cases may arise:
1. If every student gets a separate chair and no chair is left vacant, then this is a case of a
one-one onto function.
2. If every student gets a separate chair and still some chairs are left vacant, it is a oneone
into function.
3. If every student does not sit on a separate chair, i.e. more than one student sit on a
chair, no chair is left vacant, it is a many-one onto function.
Invertible Functions
The domain and range of any function may be interchanged to form a new function. It is
obtained by
interchanging the position of elements in the ordered pair of original function.
A function f : A B is invertible if its inverse relation f –1 is a function from B to A.
In general, the inverse relation f –1 may not be a function. The following theorem gives
simple criteria which tells us when it is.
Theorem: A function f: A ® B is invertible if and only if f is both one-to-one and
onto.
If f : A B is one-to-one and onto, then f is called a one-to-one correspondence
between A and B. This terminology comes from the fact that each element of A will then
correspond to a unique element of B and vice versa.
Illustrations
1. Let f : A B such that y or f(x) = x + 1, for all x A and y B be a function.
Since f is everywhere defined as well as one-one onto, therefore f is invertible. Thus f –1 is
also one-one onto function from set B to A.
2. Let f : R R such that y or f(x) =x2, for all x A and y B be a function.
Since R is the set of real numbers, where f( -2 ) = f(2) = 4, therefore f is not one-one.
Hence f is not invertible.
3. Let j : A B be a function defined everywhere. Let A and B two finite sets containing
equal number of elements. In such a case
(a) If f is one-one, then f is onto.
(b) If f is into, then f is one-one.
Moreover, when A = B we need to prove only that a function is one-one onto to show
that it is bijection.
Remarks
1. A function which possesses an inverse is said to be invertible.
2. Only one-one and onto functions are invertible.
3. The inverse function of the given function f is unique.
4. Inverse function f -1 is onto if and only if f is everywhere defined.
12.5 Composition of Functions
Let g be a function from the set A to the set B and let f be a function from the set B
to the set C. The composition of the functions f and g, denoted by f o g, is defined by
(f o g)(a) = f(g(a)).
In other words, f o g is the function that assigns to the element a of A the element
assigned by f to g(a). Note that the composition f o g cannot be defined unless the range
of g is a subset of the domain of f.
Example: Let g be the function from the set {a, b, e} to itself such that g(a) = b, g(b) = c,
and g(c)=a. Let f be the function from the set {a, b, e} to the set {I, 2, 3} such that f(a) =
3, f(b) = 2, and f(c) = 1. What is the composition of f and g, and what is the composition
of g and f?
Solution: The composition f o g is defined by (f o g) (a) = f(g(a)) = f(b) = 2, (f o g) (b) =
f(g(b)) = f(c) = 1, and (f o g) (c) = f(g(e)) = f(a) = = 3.
Note that g o f is not defined, because the range of f is not a subset of the domain of g.
12.6 Lesson End Activities
1. What is a function from X to Y?
2. Explain how to use an arrow diagram to depict a function.
3. What is the graph of a function?
4. Define one-to-one function. Give an example of a one-to-one function. Explain
how to use an arrow diagram to determine whether a function is one-to-one.
5. Define onto function. Give an example of an onto function, Explain how to use
an arrow diagram to determine whether a function is onto.
6. What is a bijection? Give an example of a bijection.
7. Define inverse function. Give an example of a function and its inverse. Given
the arrow diagram of a function, how can we find the arrow diagram of the
inverse function?
8. Define composition of functions. How is the composition of f and g denoted?
Give an example of functions f and g and their composition. Given the arrow
diagrams of two functions, how can we find the arrow diagram of the
composition of the functions?
9. What is a binary operator? Give an example of a binary operator.
10. What is a unary operator? Give an example of a unary operator.
Determine whether each relation in Exercises 11-15 is a function from X = {l,2,3,4}
to Y={a,b,c,d}. If it is a function, find its domain and range, draw its arrow diagram,
and determine if it is one-to-one or onto, if it is both one-to-one and onto, give the
description of the inverse function as a set of ordered pairs, draw its arrow diagram,
and give the domain and range of the inverse function.
11) {(1,a),(2,a),(3,c),(4,b)}
12) {(1,c),(2,a),(3,b),(4,c),(2,d)}
13) {(1,c),(2,d),(3,a),(4,b)}
14) {(1,d),(2,d),(4,a)}
15) {(1,b),(2,b),(3,b),(4,b)}
12.7 References
1) J K Sharma, ”Discrete Mathematics”
2) Kenneth H.Rosen, “Discrete Mathematics and its Applications”
3) Seymour Lipschutz and Marc Lipson, “Discrete Mathematics”
4) Charles D.Miller and Others, “Mathematical Ideas”
5) Wikipedia, the free encyclopedia.
Unit – V
Graph Theory
Contents:
13.0 Aims and objectives
13.1 Introduction
13.2 Graph
13.3 Directed Graph
13.4 Path, trail and walk
13.5 Applications of graphs
13.6 Let Us Sum Up
13.7 Lesson End Activities
13.0 Aims and objectives
There are various types of graphs, each with its own definition. Unfortunately,
some people apply the term “graph” rather loosely, so you can’t be sure what type of
graph they’re talking about unless you ask them. After you have finished this chapter, we
expect you to use the terminology carefully, not loosely.
13.1 Introduction
In mathematics and computer science, graph theory is the study of graphs;
mathematical structures used to model pairwise relations between objects from a certain
collection. The notion of a “graph” is deceptively simple: It is collection of ‘vertices’ (or
‘points’ or 'nodes') and a collection of ‘edges’ (or ‘lines’) that connect pairs of vertices. A
graph may be undirected, meaning that there is no distinction between the two vertices
associated with each edge, or its edges may be directed from one vertex to another. The
graphs studied in graph theory should not be confused with "graphs of functions" and
other kinds of graph.
13.2 Graph
What is a Graph?
There are various types of graphs, each with its own definition. To motivate the
various definitions, we’ll begin with some examples.
Example 1 (A computer network) Computers are often linked with one another so that
they can interchange information. Given a collection of computers, we would like to
describe this linkage in fairly clean terms so that we can answer questions such as “How
can we send a message from computer A to computer B using the fewest possible
intermediate computers?”
We could do this by making a list that consists of pairs of computers that are connected.
Note that these pairs are unordered since, if computer C can communicate with computer
D, then the reverse is also true. (There are sometimes exceptions to this, but they are rare
and we will assume that our collection of computers does not have such an exception.)
Also, note that we have implicitly assumed that the computers are distinguished from
each other: It is insufficient to say that “A PC is connected to a Mac.” We must specify
which PC and which Mac. Thus, each computer has a unique identifying label of some
sort.
For people who like pictures rather than lists, we can put dots on a piece of paper, one for
each computer. We label each dot with a computer’s identifying label and draw a curve
connecting two dots if and only if the corresponding computers are connected. Note that
the shape of the curve does not matter (it could be a straight line or something more
complicated) because we are only interested in whether two computers are connected or
not. Below are two such pictures of the same graph. Each computer has been labeled by
the initials of its owner.
Fig. 13.1
Computers (vertices) are indicated by dots (•) with labels. The connections (edges) are
indicated by lines. When lines cross, they should be thought of as cables that lie on top of
each other — not as cables that are joined.
In our example, the vertices are the computers and a pair of computers is in E if and only
if they are connected.
The notation Pk(V) stands for the set of all k-element subsets of the set V . Based on the
previous example, we have
Definition 1 A simple graph G is a pair G = (V,E) where
• V is a finite set, called the vertices, points, or nodes of G, and
• E is a subset of P2(V ) (i.e., a set E of two-element subsets of V ), called
the edges of G.
Vertices u and v are said to be adjacent if there is an edge e={u, v}. In such a case,
u and v are called the end points of e and e is said to be connecting u and v.
Example 2 (Routes between cities) Imagine four cities named A, B, C and D. Between
these cities there are various routes of travel, denoted by a, b, c, d, e, f and g. Here is
picture of this situation:
Fig. 13.2
Looking at this picture, we see that there are three routes between cities B and C. These
routes are named d, e and f. Our picture is intended to give us only information about the
interconnections between cities. It leaves out many aspects of the situation that might be
of interest to a traveler. For example, the nature of these routes (rough road, freeway, rail,
etc.) is not portrayed. Furthermore, unlike a typical map, no claim is made that the picture
represents in any way the distances between the cities or their geographical placement
relative to each other. The object shown in this picture is called a graph.
Following our previous example 1, one is tempted to list the pairs of cities that are
connected; in other words, to extract a simple graph from the information. Unfortunately,
this does not describe the problem adequately because there can be more than one route
connecting a pair of cities; e.g., d, e and f connecting cities B and C in the figure 13.2.
How can we deal with this? Here is a precise definition of a graph of the type required to
handle this type of problem.
Definition 2 (Graph) A graph is a triple G = (V,E, ö) where
• V is a finite set, called the vertices of G,
• E is a finite set, called the edges of G, and
• ö is a function with domain E and codomain P2(V ).
In the pictorial representation of the cities graph, G = (V, E, ö) where
V = {A,B,C,D}, E = {a, b, c, d, e, f, g}
and
ö=
Definition 2 tells us that to specify a graph G it is necessary to specify the sets V and E
and the function ö. We have just specified V and ö in set theoretic terms. The picture of
the cities graph specifies the same V and ö in pictorial terms. The set V is represented
clearly by dots (•), each of which has a city name adjacent to it. Similarly, the set E is
also represented clearly. The function ö is determined from the picture by comparing the
name attached to a route with the two cities connected by that route. Thus, the route name
d is attached to the route with endpoints B and C. This means that ö(d) = {B,C}.
The function ö is sometimes called the incidence function of the graph. The two elements
of ö(x) = {u, v}, for any x E, are called the vertices of the edge x, and we say u and v
are joined by x. We also say that u and v are adjacent vertices and that u is adjacent to v
or , equivalently, v is adjacent to u. For any v V, if v is a vertex of an edge x then we
say x is incident on v. Likewise, we say v is a member of x, v is on x, or v is in x. Of
course, v is a member of x actually means v is a member of ö(x).
Simple graphs are graphs: We can easily reconcile our two definitions by realizing that
a simple graph is a special case of a graph. Let G = (V,E) be a simple graph. Define
ö:EE to be the identity map; i.e., ö (e) = e for all e E. The graph G’ = (V,E, ö) is
essentially the same as G. There is one subtle difference in the pictures: The edges of G
are unlabeled but each edge of G’ is labeled by a set consisting of the two vertices at its
ends.
13.3 Directed Graph
A graph is said to be the directed graph if it required to associate a direction with
each edge of the graph, that is, edges are ordered pairs of distinct vertices. Before we’ll
begin to define a directed graph let us see the following example.
Example 3 (Flow of commodities) Look again at Example 2. Imagine now that the
symbols a, b, c, d, e, f and g, instead of standing for route names, stand for commodities
(bread, computers, etc.) that are produced in one town and shipped to another town. In
order to get a picture of the flow of commodities, we need to know the directions in
which they are shipped. This information is provided by picture below:
In set-theoretic terms, the information needed to construct the picture 13.3 can be
specified by giving a pair D = (V,E, ö) where ö is a function. The domain of the function
ö is E = {a, b, c, d, e, f, g} and the codomain is V × V. Specifically,
ö=
Fig. 13.3
The structure specified by this information is an example of a directed graph, which we
now define.
Definition 3 (Directed graph). A directed graph (or digraph) is a triple D = (V, E, ö)
where V and E are finite sets and ö is a function with domain E and codomain V × V. We
call E the set of edges of the digraph D and call V the set of vertices of D.
Just as with graphs, we can define a notion of a simple digraph. A simple digraph
is a pair D = (V,E), where V is a set, the vertex set, and E V × V is the edge set. Just as
with simple graphs and graphs, simple digraphs are a special case of digraphs in which ö
is the identity function on E; that is, ö(e) = e for all e E.
There is a correspondence between simple graphs and simple digraphs that is fairly
common in applications of graph theory.
Here is a picture (Fig. 13.4) of a simple graph and its corresponding digraph:
Fig. 13.4
Each, edge that is not a loop in the simple graph is replaced by two edges “in opposite
directions” in the corresponding simple digraph. A loop is replaced by a directed loop
(e.g.,{A} is replaced by (A,A)).
Indegree
The indegree of a vertex v in a directed graph is the number of edges ending at it
(or) the indegree of a vertex v is the number of edges that have v as the head. It is denoted
as indeg(v).
Outdegree
The outdegree of a vertex v in a directed graph is the number of edges beginning
from it or the outdegree of v is the number of arcs that have v as the tail. It is denoted as
outdeg(v).
Degree of a vertex
The degree of a vertex in a graph is the number of edges that touch it and is
denoted as deg(v) where v is the vertex.
Pendant (or end) vertex
A vertex is said to be pendant (or end) if its degree is one.
Isolated vertex
A vertex is said to be isolated if its degree is zero.
Even Vertex and Odd Vertex
A vertex is said to even or odd according as its degree is an even or an odd
number.
13.4 Path, trail and walk
A basic method for studying graphs and digraphs is to study substructures of these
objects and their properties. One of the most important of these substructures is called
a path.
Definition 4 (Path, trail, walk and vertex sequence) Let G = (V,E, ö) be a graph.
Let e1, e2,…, en-1 be a sequence of elements of E (edges of G) for which there is a
sequence a1, a2, . . . , an of distinct elements of V (vertices of G) such that ö(ei) = {ai, ai+1}
for i = 1, 2, . . . , n – 1. The sequence of edges e1, e2, . . ., en-1 is called a path in G. The
sequence of vertices a1, a2, . . . , an is called the vertex sequence of the path. (Note that
since the vertices are distinct, so are the edges.)
If we require that e1, e2, . . ., en-1 be distinct, but not that a1, a2, . . . , an be distinct, the
sequence of edges is called a trail.
If we do not even require that the edges be distinct, it is called a walk.
If G = (V,E, ö) is a directed graph, then ö(ei) = {ai, ai+1} is replaced by ö(ei) = (ai, ai+1) in
the above definition to obtain a directed path, trail, and walk respectively.
Note that the definition of a path requires that it does not intersect itself (i.e., have
repeated vertices), while a trail may intersect itself. Although a trail may intersect itself, it
may not have repeated edges, but a walk may. If P = (e1, e2,…, en-1) is a path in G = (V,E,
ö) with vertex sequence a1, a2, . . . , an then we say that P is a path from a1 to an. Similarly
for a trail or a walk.
In the graph of Example 2 (Fig. 13.2), the sequence c, d, g is a path with vertex
sequence A,C,B,D.If the graph is of the form G = (V,E) with E P2(V ), then the vertex
sequence alone specifies the sequence of edges and hence the path. Thus, Example 1(Fig.
13.1), the vertex sequence MN, SM, SE, TM specifies the path {MN, SM}, {SM, SE},
{SE, TM}. Similarly for digraphs. Consider the graph of Example 3 (Fig. 13.3). The edge
sequence P = (g, e, c) is a directed path with vertex sequence (D,B,C,A). The edge
sequence P = (g, e, c, b, a) is a directed trail, but not a directed path. The edge sequence P
= (d, e, d) is a directed walk, but not a directed trail.
Note that every path is a trail and every trail is a walk, but not conversely. However,
we can show that, if there is a walk between two vertices, then there is a path. This rather
obvious result can be useful in proving theorems, so we state it as a theorem.
Theorem 2 (Walk implies path) Suppose u v are vertices in the graph G = (V,E, ö).
The following are equivalent:
(a) There is a walk from u to v.
(b) There is a trail from u to v.
(c) There is a path from u to v.
Furthermore, given a walk from u to v, there is a path from u to v all of whose edges are
in the walk.
The length l of a walk is the number of edges that it uses. For an open walk, l = n–1,
where n is the number of vertices visited (a vertex is counted each time it is visited). For
a closed walk, l = n (the start/end vertex is listed twice, but is not counted twice).
Closed and open walk
A walk is closed if its first and last vertices are the same, and open if they are
different.
Fig. 13.5
Eulerian path
A path which passes through every edge (once and only once). If the starting and
ending nodes are the same, it is an Euler cycle or an Euler circuit. If the starting and
ending nodes are different, it is an Euler trail.
Hamiltonian path
A path which passes through every node once and only once. If the starting and
ending nodes are adjacent, it is a Hamiltonian cycle.
Finite graph
A finite graph has a finite number of vertices and a finite number of edges.
Observe that a graph with a finite number of vertices must automatically have a finite
number of edges and so must be finite.
Trivial graph
A finite graph with one vertex and no edges, i.e., a single point, is called the
trivial graph.
Order and Size of the graph
If G(V,E) is finite then V(G) denotes the number of vertices in G and is called the
order of G.
If G(V,E) is finite then E(G) denotes the number of edges in G and is called the
size of G.
Cycles or Loops
A loop is an edge that connects a vertex to itself. If a graph contains a cycle, it is
cyclic; otherwise it is acyclic. A directed acyclic graph is called a dag from its acronym.
In the above graph, edge e6 forms a loop.
13.5 Applications of graphs
Applications of graph theory are primarily, but not exclusively, concerned with labeled
graphs and various specializations of these.
Structures that can be represented as graphs are ubiquitous, and many problems of
practical interest can be represented by graphs. The link structure of a website could be
represented by a directed graph: the vertices are the web pages available at the website
and a directed edge from page A to page B exists if and only if A contains a link to B. A
similar approach can be taken to problems in travel, biology, computer chip design, and
many other fields. The development of algorithms to handle graphs is therefore of major
interest in computer science.
A graph structure can be extended by assigning a weight to each edge of the graph.
Graphs with weights, or weighted graph, are used to represent structures in which
pairwise connections have some numerical values. For example if a graph represents a
road network, the weights could represent the length of each road. A digraph with
weighted edges in the context of graph theory is called a network.
Networks have many uses in the practical side of graph theory, network analysis (for
example, to model and analyze traffic networks). Within network analysis, the definition
of the term "network" varies, and may often refer to a simple graph.
Many applications of graph theory exist in the form of network analysis. These split
broadly into two categories. Firstly, analysis to determine structural properties of a
network, such as the distribution of vertex degrees and the diameter of the graph. A vast
number of graph measures exist, and the production of useful ones for various domains
remains an active area of research. Secondly, analysis to find a measurable quantity
within the network, for example, for a transportation network, the level of vehicular flow
within any portion of it.
Graph theory is also used to study molecules in chemistry and physics. In condenced
matter physics, the three dimensional structure of complicated simulated atomic
structures can be studied quantitatively by gathering statistics on graph-theoretic
properties related to the topology of the atoms. For example, Franzblau's shortest-path
(SP) rings. In chemistry a graph makes a natural model for a molecule, where vertices
represent atoms and edges bonds. This approach is especially used in computer
processing of molecular structures, ranging from chemical editors to database searching.
Graph theory is also widely used in socialogy as a way, for example, to measure actor’s
prestige or to explore diffusion mechanisms, notably through the use of social network
analysis software.
13.6 Let Us Sum Up
graph, network
An abstraction of relationships among objects. Graphs consist exclusively of nodes
and edges. All characteristics of a system are either eliminated or subsumed into these
elements.
diagram, drawing
A visible rendering of the abstract concept of a graph.
point, node, vertex
Objects ("things") represented in a graph. These are almost always rendered as round
dots.
edge, link, arc
Relationships represented in a graph. These are always rendered as straight or curved
lines. The term "arc" may be misleading.
undirected
A graph in which each edge symbolizes an unordered, transitive relationship
between two nodes. Such edges are rendered as plain lines or arcs.
directed, digraph
A graph in which each edge symbolizes an ordered, non-transitive relationship
between two nodes. Such edges are rendered with an arrowhead at one end of a line
or arc.
unweighted
A graph in which all the relationships symbolized by edges are considered equivalent.
Such edges are rendered as plain lines or arcs.
weighted (edge)
Weighted edges symbolize relationships between nodes which are considered to have
some value, for instance, distance or lag time. Such edges are usually annotated by a
number or letter placed beside the edge.
weighted (vertex)
Weighted nodes have some value different from their identification.
adjacent
Two edges are adjacent if they have a node in common; two nodes are adjacent if
they have an edge in common.
degree
The number of edges which connect a node.
regular
A graph in which each node has the same degree.
complete
A graph in which every node is linked to every other node. For a complete digraph,
this means one link in either direction.
route
A sequence of edges and nodes from one node to another. Any given edge or node
might be used more than once.
path
A route that does not pass any edge more than once. If the path does not pass any
node more than once, it is a simple path.
connected
If some route exists from every node to every other, the graph is connected. Note that
some graphs are not connected. A diagram of an unconnected graph may look like
two or more unrelated diagrams, but all the nodes and edges shown are considered as
one graph.
loop, cycle
A path which ends at the node where it began.
tree
A connected graph with no loops.
Eulerian path
A path which passes through every edge (once and only once). If the starting and
ending nodes are the same, it is an Euler cycle or an Euler circuit. If the starting and
ending nodes are different, it is an Euler trail.
Hamiltonian path
A path which passes through every node once and only once. If the starting and
ending nodes are adjacent, it is a Hamiltonian cycle.
13.7 Lesson End Activities
Review Questions
1. What is a path?
2. What is a simple path?
3. Give an example of a path that is not a simple path.
4. What is a cycle?
5. What is a simple cycle?
6. Give an example of a cycle that is not a simple cycle.
7. Define connected graph.
8. Give an example of a connected graph.
9. Give an example of a graph that is not connected.
10. What is a component of a graph?
11. Give an example of a component of a graph.
12. It a graph is connected, how many components does it have?
13. Define degree of vertex v.
14. What is the relationship between the sum of the degrees of the vertices in a
graph and the number of edges in the graph.
15. In any graph, must the number of vertices of odd degree be even?
16. State a necessary and sufficient condition that a graph have a path with no
repeated edges from v to w (v w) containing all the edges and vertices.
17. If a graph G contains a cycle from v to v. must G contain a simple cycle from v
to v?
Exercises
In Exercises 1-9, tell whether the given path in the graph is
(a) A simple path (b) A cycle (c) A simple cycle
1) (b, b)
2) (e, d, c, b)
3 ) (a, d, c, d, e)
4) ( d. c. b. e, d)
5) (b, c, d, a, b, e, d, c, b)
6) (a, d, c, h, e)
7) (b, c, d, e, b, b)
8) (d)
9) (d, c, b)
Subgraphs, Types of Graphs and Graph Representations.
Contents
14.0 Aims and Objectives
14.1 Introduction
14.2 Subgraphs
14.3 Types of Graphs
14.4 Representation of graphs in computer memory
14.4.1 List structures
14.4.2 Matrix structures
14.5 Chapter Summary
14.6 Lesson End Activities
14.7 References
14.0 Aims and Objectives
The main aim and objective of this chapter is know about the subgraphs, various
types of graphs and the different ways of representing the graphs in a computer.
After reading this unit, you should be able to
Construct a subgraph from a graph.
Know about the various types of graphs
Know about the ways of representing the graphs in a computer memory.
14.1 Introduction
In this lesson, we first discuss about the subgraphs. A new graph can obtained by
selecting certain edges and vertices from a given graph with certain restrictions. The
graphs so obtained are called subgraphs. Next we discuss about the types of graphs. We
conclude the lesson by discussing the various methods available for representing a graph
in computers.
14.2 Subgraphs
A subgraph G’ of a graph G is obtained by selecting certain edges and vertices
from G subject to the restriction that if we select an edge e in G that is incident on
vertices v and w, we must include v and w in G’. The restriction is to ensure that G’ is
actually a graph. The formal definition follows.
Definition 1 (Subgraph) Let G = (V,E,ö) be a graph. A graph G' = (V',E',ö') is a
subgraph of G if V' V , E' E, and ö' is the restriction of ö to E'.
As we have noted, the fact that G' is itself a graph means that ö(x) P2(V’ ) for each X 
E' and, in fact, the codomain of ö' must be P2(V’). If G is a graph with loops, the
codomain of ö' must be P2(V') P 1(V'). This definition works equally well if G is a
digraph. In that case, the codomain of ö' must be V × V.
Example 1 (Subgraph — key information) For the graph G = (V,E,ö) below, let
G'=(V',E', ö') be defined by V' = {A,B,C}, E' = {a, b, c, f}, and by ö' being the restriction
of ö to E' with codomain P2(V'). Notice that ö' is determined completely from knowing
V', E' and ö. Thus, to specify a subgraph G', the key information is V' and E'.
As another example from the same graph, we let V' = V and E' = {a, b, c, f}. In this case,
the vertex D is not a member of any edge of the subgraph. Such a vertex is called an
isolated vertex of G'.
One way of specifying a subgraph is to give a set of edges E' E and take V' to be
the set of all vertices on some edge of E'. In other words, V' is the union of the sets ö(x)
over all x. E'. Such a subgraph is called the subgraph induced by the edge set E' or the
edge induced subgraph of E'. The first subgraph of this example is the subgraph induced
by E' = {a, b, c, f}.
Fig. 14.1
Likewise, given a set V' V, we can take E' to be the set of all edges x E such that
ö(x) V'. The resulting subgraph is called the subgraph induced by V ' or the vertex
induced subgraph of V'. Referring to the picture again, the edges of the subgraph induced
by V' = {C,B}, are E' = {d, e, f}. Look again at the above graph. In particular, consider
the path c, a with vertex sequence C,A,B. Notice that the edge d has ö(d) = {C,B}. The
subgraph G' = (V ',E', ö'), where V ' = {C,A,B} and E' = {c, a, d} is called a cycle of G.
In general, whenever there is a path in G, say e1, . . . , en-1 with vertex sequence a1, . . . ,
an, and an edge x with ö(x) = {a1, an}, then the subgraph induced by the edges e1, . . . , en1, x is called a cycle of G. Parallel edges like a and b in the preceding figure induce a
cycle. A loop also induces a cycle.
The formal definition of a cycle is:
Definition 2 (Circuit and Cycle) Let G = (V,E, ö) be a graph and let e1, . . . , en be a trail
with vertex sequence a1, . . . , an, a1. (It returns to its starting point.) The subgraph G' of G
induced by the set of edges {e1, . . . , en} is called a circuit of G. The length of the circuit
is n.
• If the only repeated vertices on the trail are a1 (the start and end), then the
circuit is
called a simple circuit or cycle.
• If “trail” is replaced by directed trail, we obtain a directed circuit and a directed
cycle.
In our definitions, a path is a sequence of edges but a cycle is a subgraph of G. In actual
practice, people often think of a cycle as a path, except that it starts and ends at the same
vertex. This sloppiness rarely causes trouble, but can lead to problems in formal proofs.
Cycles are closely related to the existence of multiple paths between vertices.
14.3 Types of Graphs
Null Graph
The simplest type of graph is a null graph. It consists of a non-empty finite set of
elements called vertices.
Complete Graph
A complete graph is a graph where every pair of distinct vertices are adjacent i.e., there is
an edge between every pair of distinct vertices. A complete graph on n vertices is denoted
by Kn (or sometimes by K(n) ). So, for example, figure 14.2 is the graph K5.
Fig. 14.2, The Graph K5
Regular Graph
A graph in which every vertex has the same degree is called a regular graph. If every
vertex has degree r then we say the graph is regular of degree r. All null graphs are
regular of degree zero.
Weighted Graph
A graph with numbers on the edges is called a weighted graph. If edge e is labeled
k, we say that the weight of edge e is k. In a weighted graph, the length of a path is the
sum of the weights of the edges in the path.
Bipartite Graph
A graph G=(V, E) is bipartite if there exist subsets V1 and V2(either possibly
empty) of V such that V1 V2= , V1 V2= V and each edge in E is incident on one
vertex in V1 and one vertex in V2.
Example
The graph in fig. is bipartite since if we let
V1={v1,v2,v3} and V2={v4,v5},
Each edge is incident on one vertex in V1 and one vertex in V2.
Fig. 14.3
Infinite Graph
A graph is infinite if it has infinitely many vertices or edges or both; otherwise
the graph is finite. An infinite graph where every vertex has finite degree is called locally
finite. When stated without any qualification, a graph is usually assumed to be finite.
14.4 Representation of graphs in computer memory
There are different ways to store graphs in a computer system. The data structure used
depends on both the graph structure and the algorithm used for manipulating the graph.
Theoretically one can distinguish between list and matrix structures but in concrete
applications the best structure is often a combination of both. List structures are often
preferred for sparse graphs as they have smaller memory requirements. Matrix structures
on the other hand provide faster access for some applications but can consume huge
amounts of memory.
14.4.1 List structures
Incidence List
The edges are represented by an array containing pairs (ordered if directed) of
vertices (that the edge connects) and possibly weight and other data.
Adjacency List
Much like the incidence list, each vertex has a list of which vertices it is adjacent to.
This causes redundancy in an undirected graph: for example, if vertices A and B are
adjacent, A's adjacency list contains B, while B's list contains A. Adjacency queries
are faster, at the cost of extra storage space.
Example
Fig. 14.4
The graph pictured above (Fig. 14.4) has this adjacency list representation:
a adjacent to b,c
b adjacent to a,c
c adjacent to a,b
14.4.2 Matrix structures
Incidence matrix
The graph is represented by a matrix of E (edges) by V (vertices), where [edge,
vertex] contains the edge's data (simplest case: 1 - connected, 0 - not connected).
Adjacency matrix
There is an n by n matrix, where n is the number of vertices in the graph. If there is an
edge from some vertex ‘x’ to some vertex ‘y’, then the element Mx,y is 1, otherwise it
is 0. This makes it easier to find subgraphs, and to reverse graphs if needed.
Examples
Here is a simple example of a labeled graph and its adjacency matrix. The convention
followed here is that an adjacent edge counts 1 in the matrix for an undirected graph.
Labeled Graph Adjacency matrix
The adjacency matrix of a complete graph is all 1's except for 0's on the diagonal.
Laplacian matrix or Kirchhoff matrix or Admittance matrix
Is defined as degree matrix minus adjacency matrix and thus contains adjacency
information and degree information about the vertices.
Distance matrix
A symmetric n by n matrix an element Mx,y of which is the length of shortest path
between x and y; if there is no such path Mx,y = infinity. It can be derived from
powers of the Adjacency matrix.
14.5 Chapter Summary
Subgraph
A graph G’ whose vertices and edges
form subsets of the vertices and edges of a given graph
G . If G’
is a subgraph of
G , then G
is said to be a supergraph of
G’ .
Null Graph
It consists of a non-empty finite set of elements called vertices.
Complete Graph
A complete graph is a graph where every pair of distinct vertices are
adjacent i.e., there is an edge between every pair of distinct vertices. A complete
graph on n vertices is denoted by Kn (or sometimes by K(n) ).
Regular Graph
A graph in which every vertex has the same degree is called a regular graph.
Weighted Graph
A graph with numbers on the edges is called a weighted graph. In a weighted graph,
the length of a path is the sum of the weights of the edges in the path.
Bipartite Graph
A graph G=(V, E) is bipartite if there exist subsets V1 and V2(either possibly
empty) of V such that V1 V2= , V1 V2= V and each edge in E is incident on
one vertex in V1 and one vertex in V2.
Infinite Graph
A graph is infinite if it has infinitely many vertices or edges or both; otherwise the
graph is finite.
List structures
Incidence List
The edges are represented by an array containing pairs (ordered if directed) of
vertices (that the edge connects) and possibly weight and other data.
Adjacency List
Much like the incidence list, each vertex has a list of which vertices it is
adjacent to.
Matrix structures
Incidence matrix
The graph is represented by a matrix of E (edges) by V (vertices), where [edge,
vertex] contains the edge's data (simplest case: 1 - connected, 0 - not connected).
Adjacency matrix
There is an n by n matrix, where n is the number of vertices in the graph. If there is
an edge from some vertex ‘x’ to some vertex ‘y’, then the element Mx,y is 1,
otherwise it is 0. This makes it easier to find subgraphs, and to reverse graphs if
needed.
The adjacency matrix of a complete graph is all 1's except for 0's on the diagonal.
Laplacian matrix or Kirchhoff matrix or Admittance matrix
Is defined as degree matrix minus adjacency matrix and thus contains adjacency
information and degree information about the vertices.
Distance matrix
A symmetric n by n matrix an element Mx,y of which is the length of shortest path
between x and y; if there is no such path Mx,y = infinity. It can be derived from
powers of the Adjacency matrix.
14.6 Lesson End Activities
Review Questions
1. What is a subgraph?
2. Give an example of a subgraph.
In Exercises 1-5, write the adjacency matrix of each graph.
4. The complete graph on five vertices K5
5. The complete bipartite graph K2.3
In Exercises 6-8, write the incidence matrix of each graph.
6. The graph of Exercise 1
7. The graph of Exercise 2
8. The graph of Exercise 3
In Exercises 13-16, draw the graph represented by each adjacency matrix.
17) Write the adjacency matrices of the components of the graphs given by the
adjacency matrices of Exercises 13-16.
In Exercises 24 and 25, draw the graphs represented by the incidence matrices.
1. What is an adjacency matrix?
2. If A is the adjacency matrix of a simple graph, what are the values of the entries in An?
3. What is an incidence matrix?
14.7 References
1) J K Sharma, ”Discrete Mathematics”
2) Kenneth H.Rosen, “Discrete Mathematics and its Applications”
3) Seymour Lipschutz and Marc Lipson, “Discrete Mathematics”
4) Charles D.Miller and Others, “Mathematical Ideas”
5) Wikipedia, the free encyclopedia.
Trees
Contents
15.0 Aims and objectives
15.1 Introduction
15.2 Tree
15.2.1 Rooted tree
15.2.2 Other trees
15.2.3 Height of the Tree
15.2.4 Facts and Enumeration
15.3 Binary tree
15.4 Tree traversal
15.4.1 Traversal methods
15.4.2 Sample implementations
15.5 Computer Representation of Trees
15.5.1 Computer Representation of General Trees
15.5.2 Computer Representation of Binary Trees
15.6 Let Us Sum Up
15.7 Lesson End Activities
15.8 References
15.0 Aims and objectives
The main aim and objective of this lesson is know about the trees, its properties,
binary trees, tree traversing and representation of trees in computer.
After reading this unit, you should be able to
Construct a tree.
Perform inorder, preorder and postorder tree traversing
Know about the ways of representing the trees in a computer memory.
15.1 Introduction
Trees play an important role in a variety of algorithms. Trees are widely used in
computer science data structures such as binary search trees, heaps, tries, etc. In this
section, we define trees precisely and look at some of their properties.
15.2 Tree
In graph theory, a tree is a graph in which any two vertices are connected by exactly one
path. Alternatively, any connected graph with no cycles is a tree. A forest is a disjoint
union of trees.
Definition (Tree)
If G is a connected graph without any cycles then G is called a tree. (If |V| = 1, then G is
connected and hence is a tree.) A tree is also called a free tree.
The graph in fig. 15.1 is connected but is not a tree. It has many cycles, including ({A,
B, C}, {a, e, c}). The subgraph of this graph induced by the edges {a, e, g} is a tree. If
G is a tree, then ö is an injection since if e1 e2 and ö(e1) = ö(e2), then {e1, e2} induces a
cycle. In other words, any graph with parallel edges is not as tree. Likewise, a loop is a
cycle, so a tree has no loops. Thus, we can think of a tree as a simple graph when we are
not interested in names of the edges.
Since the notion of a tree is so important, it will be useful to have some equivalent
definitions of a tree. We state them as a theorem
Theorem 1 (Alternative definitions of a tree) If G is a connected graph, the
following are equivalent.
(a) G is a tree.
(b) G has no cycles.
(c) For every pair of vertices u v in G, there is exactly one path from u to v.
(d) Removing any edge from G gives a graph which is not connected.
(e) The number of vertices of G is one more than the number of edges of G.
A directed tree is a directed graph which would be a tree if the directions on the edges
were ignored.
Example 1:
The example tree shown in fig. 15.2 has 6 vertices and 6 - 1 = 5 edges. The unique
simple path connecting the vertices 2 and 6 is 2-4-5-6.
15.2.1 Rooted tree
A tree is called a rooted tree if one vertex has been designated the root, in which case
the edges have a natural orientation, towards or away from the root.
Definitions for rooted trees
A directed edge refers to the link from the parent to the child (the arrows in the
picture of the tree).
The root node of a tree is the node with no parents. There is at most one root node
in a rooted tree.
A leaf is a node that has no children.
The depth of a node n is the length of the path from the root to the node. The set
of all nodes at a given depth is sometimes called a level of the tree. The root node
is at depth zero.
The height of a tree is the depth of its furthest leaf. A tree with only a root node
has a height of zero.
Siblings are nodes that share the same parent node.
If a path exists from node p to node q, where node p is closer to the root node than
q, then p is an ancestor of q and q is a descendant of p.
The size of a node is the number of descendants it has including itself.
Rooted trees, often with additional structure such as ordering of the neighbors at each
vertex, are a key data structure in computer science. In a context where trees are
supposed to have a root, a tree without any designated root is called a free tree.
A polytree has at most one undirected path between any two vertices. In other words, a
polytree is a directed acyclic graph (DAG) for which there is no undirected cycles either.
15.2.2 Other trees
A labeled tree is a tree in which each vertex is given a unique label. The vertices of a
labeled tree on n vertices are typically given the labels 1, 2, …, n.
An irreducible (or series-reduced) tree is a tree in which there is no vertex of degree 2.
An ordered tree is a tree for which an ordering is specified for the children of each node.
15.2.3 Height of the Tree
The root of the tree has level 0, and the level of any other vertex in the tree is one
more than the level of its parent. In Fig.15.3, vertices v11, v12, v21, v22, v31 and v32 are at
level. The depth (or height) of a tree is the maximum level of any leaf in the tree. This
equals the length of the longest path from the root of any leaf. The tree in Fig.15.3 is of
height 2.
Theorem 2 There is unique path between each pair of vertices in a tree.
Theorem 3 A tree with n vertices has n - 1 edges.
15.2.4 Facts and Enumeration
Facts
Every tree is a bipartite graph. Every tree with only countably many vertices is a
planar graph.
Every connected graph G admits a spanning tree, which is a tree that contains
every vertex of G and whose edges are edges of G. Every connected graph even
admits a normal spanning tree.
Every non-null tree has at least one leaf, or vertex of degree 1 (If it has a vertex,
it has a leaf).
Enumeration
Given n labeled vertices, there are nn- 2 different ways to connect them to make a
tree. This result is called Cayley’s formula. It can be proved by first showing that the
number of trees with n vertices of degree d1,d2,...,dn is the multinomial coefficient
15.3 Binary tree
In computer science, a binary tree is a tree data structure in which each node has at most
two children. Typically the child nodes are called left and right. One common use of
binary trees is binary search tree; another is binary heaps.
Types of binary trees
A rooted binary tree is a rooted tree in which every node has at most two
children.
A full binary tree, or proper binary tree, is a tree in which every node has zero
or two children.
A perfect binary tree (sometimes complete binary tree) is a full binary tree in
which all leaves are at the same depth.
A complete binary tree is a tree with n levels, where for each level d <= n - 1,
the number of existing nodes at level d is equal to 2d. This means all possible
nodes exist at these levels. An additional requirement for a complete binary tree is
that for the nth level, while every node does not have to exist, the nodes that do
exist must fill from left to right. (This is ambiguous with perfect binary tree.)
A balanced binary tree is where the depth of all the leaves differs by at most 1.
A rooted complete binary tree can be identified with a free magma.
An almost complete binary tree is a tree in which each node that has a right
child also has a left child. Having a left child does not require a node to have a
right child. Stated alternately, an almost complete binary tree is a tree where for a
right child, there is always a left child, but for a left child there may not be a right
child.
A degenerate tree is a tree where for each parent node, there is only one
associated child node. This means that in a performance measurement, the tree
will behave like a linked list data structure.
The number of nodes n in a complete binary tree can be found using this
formula: n = 2h + 1 - 1 where h is the height of the tree.
The number of leaf nodes n in a complete binary tree can be found using this
formula: n = 2h where h is the height of the tree.
15.4 Tree traversal
In computer science, tree-traversal refers to the process of visiting each node in a tree
data structure, exactly once, in a systematic way. Such traversals are classified by the
order in which the nodes are visited. The following methods or algorithms are described
for a binary tree, but they may be generalized to other trees as well.
15.4.1 Traversal methods
Compared to linear data structures like linked lists and one dimensional arrays, which
have only one logical means of traversal, tree structures can be traversed in many
different ways. Starting at the root of a binary tree, there are three main steps that can be
performed and the order in which they are performed define the traversal type. These
steps are: Performing an action on the current node(referred to as “visiting” the node); or
repeating the process with the subtrees rooted at our left and right children. Thus the
process is most easily described through recursion.
To traverse a non-empty binary tree T with root R in preorder, perform the following
operations:
1. Visit the root.
2. Traverse the left subtree of R in preorder.
3. Traverse the right subtree of R in preorder.
To traverse a non-empty binary tree T with root R in inorder, perform the following
operations:
1. Traverse the left subtree of R in inorder.
2. Visit the root.
3. Traverse the right subtree of R in inorder.
To traverse a non-empty binary tree T with root R in postorder, perform the following
operations:
1. Traverse the left subtree of R in postorder.
2. Traverse the right subtree of R in postorder.
3. Visit the root.
(This is also called Depth-first traversal.)
Finally, trees can also be traversed in level-order, where we visit every node on a level
before going to a lower level. This is also called Breadth-first traversal.
Traversal where levels are visited successively, starting with level 0 (the root node), and nodes are visited
from left to right on each level.
Example 2
In this binary search tree,
Preorder traversal sequence: F, B, A, D, C, E, G, I, H
Inorder traversal sequence: A, B, C, D, E, F, G, H, I
Note that the inorder traversal of a binary search tree yields an ordered list
Postorder traversal sequence: A, C, E, D, B, H, I, G, F
Level-order traversal sequence: F, B, G, A, D, I, C, E, H
15.4.2 Sample implementations
preorder(node)
print node.value
if node.left null then preorder(node.left)
if node.right null then preorder(node.right)
inorder(node)
if node.left null then inorder(node.left)
print node.value
if node.right null then inorder(node.right)
postorder(node)
if node.left null then postorder(node.left)
if node.right null then postorder(node.right)
print node.value
All three sample implementations will require stack space proportional to the height of
the tree. In a poorly balanced tree, this can be quite considerable.
15.5 Computer Representation of trees
15.5.1 Computer Representation of General Trees
Suppose T is a general tree. The tree T will be maintained in memory by means of a
linked representation which uses three parallel arrays INFO, CHILD(or DOWN), and
SIBL(or HORZ), and a pointer variable ROOT as follows. First of all, each node N of T
will correspond to a location K such that:
(1) INFO[K] contains the data at node N.
(2) CHILD[K] contains the location of the first child of N. The condition
CHILD[K]=NULL indicates that N has no children.
(3) SIBL[K] contains the location of the next sibling of N. The condition
SIBL[K]=NULL indicates that N is last child of its parent.
Furthermore, ROOT will contain the location of the root R of T. Although this
representation may seem artificial, it has the important advantage that each node N of T,
regardless of the number of children of N, will contain exactly three fields.
The above representation may be extended to represent a forest F consisting of
trees T1, T2,…, Tm by assuming the roots of the tree are siblings. In such a case, ROOT
will contain the location of the root R1 of the tree T1; or when F is empty, ROOT will
equal NULL.
15.5.2 Computer Representation of Binary Trees
Binary trees can be constructed from programming language primitives in several
ways. In a language with records and references, binary trees are typically constructed by
having a tree node structure, which contains some data and references to its left child and
its right child. Sometimes it also contains a reference to its unique parent. If a node has
fewer than two children, some of the child pointers may be set to a special null value, or
to a special sentinel node.
Binary trees can also be stored as an implicit data structure in arrays, and if the
tree is a complete binary tree, this method wastes no space. In this compact arrangement,
if a node has an index i, its children are found at indices 2i + 1 and 2i + 2, while its parent
(if any) is found at index ((i-1)/2)(assuming
the root has index zero). This method benefits from more compact storage and better
locality of reference, particularly during a preorder traversal. However, it is expensive to
grow and wastes space proportional to 2h - n for a tree of height h with n nodes.
15.6 Let Us Sum Up
A tree is a graph in which any two vertices are connected by exactly one path.
A tree is called a rooted tree if one vertex has been designated the root.
A directed edge refers to the link from the parent to the child (the arrows in the
picture of the tree).
The root node of a tree is the node with no parents. There is at most one root node
in a rooted tree.
A leaf is a node that has no children.
The depth of a node n is the length of the path from the root to the node. The set of
all nodes at a given depth is sometimes called a level of the tree. The root node is at
depth zero.
The height of a tree is the depth of its furthest leaf. A tree with only a root node has
a height of zero.
Siblings are nodes that share the same parent node.
If a path exists from node p to node q, where node p is closer to the root node than
q, then p is an ancestor of q and q is a descendant of p.
The size of a node is the number of descendants it has including itself.
A labeled tree is a tree in which each vertex is given a unique label. The vertices of
a labeled tree on n vertices are typically given the labels 1, 2, …, n.
An irreducible (or series-reduced) tree is a tree in which there is no vertex of
degree 2.
An ordered tree is a tree for which an ordering is specified for the children of each
node.
A binary tree is a tree data structure in which each node has at most two children.
Typically the child nodes are called left and right.
A tree-traversal refers to the process of visiting each node in a tree data structure,
exactly once, in a systematic way.
15.7 Lesson End Activities
Review Questions
1. Define free tree.
2. Define rooted tree.
3. What is the level of a vertex in a rooted tree?
4. What is the height of a vertex in a rooted tree?
5. Give an example of a hierarchical definition tree.
6. What is preorder traversal?
7. Give an algorithm to execute a preorder traversal.
8. What is inorder traversal?
9. Give an algorithm to execute an inorder traversal.
10. What is postorder traversal?
11. Give an algorithm to execute a postarder traversal.
12. Explain how a tree can be used to represent an expression.
Exercises
I) In Exercises 1-5, list the order in which the vertices are processed using preorder,
inorder, and postorder traversal.
II) Which of the graphs in Exercises 1-4 are trees? Explain.
15.8 References
1) J K Sharma, ”Discrete Mathematics”
2) Kenneth H.Rosen, “Discrete Mathematics and its Applications”
3) Seymour Lipschutz and Marc Lipson, “Discrete Mathematics”
4) Charles D.Miller and Others, “Mathematical Ideas”
5) Wikipedia, the free encyclopedia.
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