Download I.2.2.Operations on sets

MOUNT KENYA UNIVERSITY
KIGALI CAMPUS
COURSE UNIT BBM/SBC 112: FOUNDATION OF MATHEMATICS
BY LECTURER CELESTIN
SEPTEMBER-DECEMBER 2013
HANDOUT ONE: SET THOERY
COURSE UNIT BBM/SBC112: FOUNDATION MATHEMATICS
SPECIFIC OBJECTIVES OF THE LESSON:
By the end of the course unit the learners should be able to:
i.
ii.
iii.
iv.
Define a set and the set concept ,
Use Venn diagrams to illustrate sets
Perform set operations
Solve business problems using set operations
COURESE CONTENT:
CHAP.ONE: SET THEORY
I.1.Definitions
I.1.1.Sets
I.1.2.Elements
I.1.3. Null set
I.2.1.Symbols used in set theory
I.2.2.Operations on sets
I.3.1.Properties of operations on sets
I.3.2.Set enumeration
I.3.3.Representation of set by the use of Venn diagrams
I.3.4 Application of set theory in business
CHA.TWO: SETS OF NUMBERS
II.1.Set of real natural numbers (N)
II.2.2.Set of prime integers (Z)
II.2.3. Set of Rational numbers (Q)
II.2.4.Set of Irrational numbers (I)
II.2.5. Set of Real numbers (  )
II.2.6.Laws of arithmetic including comparison with terms of set theory
and logic
II.2.7. Evenness, Oddness, and Denseness
II.2.8. Equations in sets of numbers
II.2.9. Inequations in sets of numbers
II.2.10.The set of complex numbers (C)
CHAP. THREE: ELEMENTARY MATHEMATICS
III.3.1.Indices and Logarithms
III.3.2.Application of indices and logarithms
III.3.3.Remainder theorem and its applications
III.3.4.Binomial expansions and its applications in business
III.3.5.Permutations and combinations functions
III.3.6.Graphs identities and Equations
CHAP. FOUR: MATRIX ALGEBRA
IV.4.1.Matrices and linear equations
IV.4.2.Vectors and matrices and their Algebra
IV.4.3.Systems of linear equations and their expressions in matrix form
IV4.4.Inverse matrices and some of its applications
IV.4.5.Application of simultaneous equations in business
IV.4.6.Solving systems of linear equations using row operations (in the
case where there is a unique solution)
IV.4.7.Transition matrices
IV.4.8.Some Economic/Managerial applications of linear equations
CHAP.FIVE: SEQUENCES AND SERIES.
V.5.1.Arthmetic and Geometric progressions
V.5.2.Some Financial application of sequences and series
Recommended Text Books:
i.
Thomas Bending (2007);Foundation mathematics and statistics;
cengage learning college
ii.
Thomas G. and Finney R (2003); Calculus; Pearson Education Inc.
Text-Books for further reading
i.
Edwards H.C (2001), Elementary Linear Algebra; Prentice Hall
Course Assessment:
Examination-70%,
Continuous Assessment Tests-30% (2 sitting CATs-20% +Assignement5% + Attendance-5%)
Total Marks-100%
Pass Marks-40%
Lecturer: Mbonigaba Celestin.
HANDOUT ONE: SETS THEORY
I.1.Definition:
A set is a collection of objects which have something in common
Eg: A is the set of days of the week.
Set theory is very useful tool in Mathematics, and gives us a “Shorthand” way of
describing things and situations which might otherwise be complicated to explain.
A set is always described by a capital letter.
I.1.2. Element of a set.
An element or a member of a set, is an object of the set, and is represented by
letters in lower cases, the members of a set are written in curly brackets.
Eg: The set of vowels, V= a, e, i, o, u , the set of the prime numbers less than 10 P=
2,3,5,7
I.1.3.The Universal set
A Universal set is a set of all the possible subsets from any consideration and is
represented by the symbol” Ƹ”.
Eg: A set of natural numbers N= 1,2,3,4,... In other words a Universal set is a set
which contains all things that we are considering.
I.1.4.Null or Empty set
The Empty or Null set is the set which has no members and is written as  or  .
We do not write 0, since this is the set with one member zero.
Eg: A set of blue tigers.
Note: An Empty set is defined to a subset of all sets. Sets can be either finite or
infinite.
i.
Finite set: A set is said to finite if the elements of the set can be
counted; there may be many elements, but they can still be counted.
Eg: F= {factors of 12}
Some sets do not have many elements, others are very big, and would be difficult
to list them.
Eg: K= {odd numbers less than 1000}, K= 1,3,5,...,999
We write the first few members of each sequence to show what the relation is,
then a series of three dots before ending with the last member.
ii.
Infinite sets: These are sets with an unlimited number of members.
Eg: W= {whole numbers}
R= {real numbers}
T= {multiples of 3}, we can also write T as 3,6,9,12,... we show that it is an
infinite set by writing the dots without a number after them.
I.2.1.Symbols used in set theory
Language is essential part of the learning of Mathematics. There are similarities
between the everyday language we use and language of Mathematics. For
example we often use symbols to convey ideas or messages. Symbols we use in
sets theory or in Mathematics in general are used to write Mathematics precisely
and concisely. The incorrect use of symbols and language can lead to the incorrect
use of Mathematics.
In set theory we use;
i.
ii.
“n(A)” to mean, the number of elements of set A. Eg: A = 1,3,5,7,9,11,
n(A) =6
“=” to mean equals. Eg: A= x, y, z and B= x, z, y, therefore A=B as the
order of their members does not matter. If they have do not have
exactly the same members, then A≠B.
iii.
iv.
v.
“  ” for is a subset of. Eg: X= 1,2,3 and Y= 1,2,3,4,5 as all the elements of
X are contained in Y then X is a subset of Y and is written as X  Y .
“  ” for is a member of. Eg: F= { 1,2,}, therefore, 1  F, 2 F.
“  ” for is not a member of.
Note: There are some other symbols used in set theory that we shall discuss when
doing exercises.
I.2.2.Operations on sets
In set theory, we normally perform three fundamental operations called, the
Union”  ”, the intersection”  ” and complement of a set A written as ( A ).The
rules for the use of such operations was developed by A. De Morgan.
1. Union of sets: If we have two sets A and B , the union of A and B is written
as A  B and is defined as a set of all elements which are members of either
set. They include all the elements which are in both sets.
Eg: if A= {2,4,6,8} and B={ 1,3,5,7}, then A  B={1,2,3,4,5,6,7,8}
2. Intersection of sets: The intersection of two sets A and B is the set of
elements that occur in both sets.
Eg: If A={ 1,3,5,15} and B ={1,2,3,4,6,12}, then A  B={1,2,3}
Note: If A  B =  the two sets are said to be disjoint.
3. The Complement of a set: If a set A is part of a Universal set, then all of th
elements of the Universal set which are not in A form the Complement of A
in the Universal set, and we write it as A .
Eg: Ƹ= 1,2,3,4,5 and {A=2, 4} then the complement of A, A  1,3,5
SET NOTATION:
There is an important piece of notation which we must understand, though we
can write down a set as:
A= {X: x ϵR, 0  x  10 }, by this we mean the set of all values of X such that x is a
real number and x is greater than 0 but less than 10. Therefore, A = {1,…, 9}
Eg: Describe the members of these sets.
i.
ii.
iii.
iv.
A= {a: a ϵZ, -2  a  3 }
B={ b: b ϵW,  9 }
C= {c: c ϵN, 2c+1  10 }
D= {d: d ϵZ, -2  d  7 }
The above operations including symbols can permit the calculations by respecting
certain rules.
1.
2.
3.
4.
5.
( A)   A
A ( A  A)  ( A  A)  A
A  ( A)   , A  ( A)  
A    A, A    A
A  B  B  A, commutativity of the Union
A  B  B  A, commutativity Of the intersection
6. A  ( A  B), ( A  B)  A
7. ( A  B)   A  B , ( A  B)   A  B 
8.
A  B  A  B   A  B  B
 A B  A
9. A  B    A  B   B  A
10. A  B    A  B  B  A
11. A  ( B  C )  ( A  B)  A  C ), orA  ( B  C )  ( A  B)  ( A  C ), distributivity Of
the Union over the intersection or, the distributivity of the intersection over
the Union respectively.
12.
A  ( B  C )  ( A  B)  C , or
A  ( B  C )  ( A  B)  C
Associativity of the intersection or the
associativity of the union respectively, i.e the order of operation does not
matter.
13.If ( A  B), then, ( A  C )  ( B  C ), and , ( A  C )  ( B  C ) .
14. (C  A, and , C  B)  C  ( A  B), ( A  C, andB  C)  A  (B  C)
I.3.1.Properties of set enumeration.
I.3.2.Set enumeration
We have seen that an empty set is a subset of every set, and a set is subset of
itself. One can ask how many subsets do a set a have. Looking at the set E= {a, b,
c} ,we can write down the following subsets of E as follows:
{ , a, b, c, a, b, a, c, b, c, a, b, c}, Which are the subsets of E, and this can be
written as PE which means subsets of E, so there are 8 subsets in all. This number
of subsets of a given set can be found by 2n subsets, where n-is the number of
elements in that set. The number of the subsets then written as n(PE)=2n.
Number of the Union of two sets A and B is given by:
i.
n( A  B)  n( A)  n( B)  n( A  B) .Then for any three sets A,B and C this
can be as follows:
ii.
n( A  B  C )  n( A)  n( B)  n(C )  n( A  B)  n( A  C )  n( B  C )  n( A  B  C )
I.3.3.Representation of set by the use of Venn diagrams
Sets are pictorially represented using Venn diagram, we usually use rectangle for
the Universal set, and show the sets using circles or ellipses inside the rectangle
but no reason why other shapes cannot be used.
Illustration:
Let A= 1,2,3,4,5, andB  2,4,6,8, this can be shown in the Venn diagram as follows:
SOLVING PROBLEMS:
i.
Problems involving two sets:
We are often given problems which look very difficult to solve, but the use
of the set theory and Venn diagrams comes to our rescue.
Worked example:
In a class of of 26 students, five say they like neither Mathematics nor Economics.
Eight like Economics but not Mathematics. Six like Mathematics but not
Economics.
a. How many like Economics or Mathematics?
b. How many like Economics and Mathematics?
Solution:
Assign the following:  = {students in the class}
E = {students who like Economics}
M= {students who like Mathematics}
Now, n (  ) =26, the number of students in the class
n( E  M )  =5 number of students who like neither subject
n( M  M ) =8 number of students who like Economics but not Mathematics
n( M  E ) =6 number of students who like Mathematics but not Economics.
Since the number of students in the class is 26 and only five like neither subject,
we can see that
(26-5)=21 like Mathematics or Economics. Therefore
n( E  M )  21, and , n( E  M )  n( E  M )  n( E )  n( M ) , so n( E  M )  7
a. Those who like Economics or Mathematics are 21 as E and M are liked by
the connector “or”
b. Those who like Economics and Mathematics are 7 as E and M are
connected by “and”
ii.
Problems involving three sets:
These are solved by using the interpretation of all the 8 regions that are
formed by three sets as follows:
In the diagram above, the Universal set is set of boys in a school.
F represents a set of boys who play football, V represents a set of boys who play
volleyball, and T represents a set of boys who play table tennis.
All of these regions can be described using word as follows:
i.
ii.
region 1- is the number of boys who like all three games, in set language
this is: n( F  T  V )
region two- is the number of boys who like Table tennis and Volleyball
but not Football, in set language this is: n( F   T  V )
iii.
iv.
region 5- is the number of boys who football only, in set language this is
: n( F  T   V )
Region 8- is the number of boys who like none of the three games and in
set language this is: n( F   T   V )or , n( F  T  V )  and so on.
Worked example:
In a village, there are 120 families. Every family grows at least one of the crops,
Beans, Yams and Cassava. 5families grow all three, 20families grow Beans and
Cassava; 30 families grow Beans and Yams; 40fmailies grow Yams and Cassava.62
families grow Beans; 18 families grow Yams only. How many families grow
Cassava only?
Solution:
I.3.4. Application of set theory in business
Introduction
Set theory is used in almost every discipline including engineering, business, and
medical and related health sciences, along with the natural sciences. In business
operations, it can be applied at every level where intersecting and nonintersecting sets are identified. For example, the sets for warehouse operations
and sales operations are both intersected by the inventory set. To improve the
cost of goods sold, the solution might be found by examining where inventory
intersects both sales and warehouse operations
Business Applications
Applied to business operations, set theory can assist in planning and operations.
Every element of business can be grouped into at least one set such as
accounting, management, operations, production and sales. Within those sets are
other sets. In operations, for example, there are sets of warehouse operations,
sales operations and administrative operations. In some cases, sets intersect -- as
sales operations can intersect the operations set and the sales set
Rough-Set Theory
Rough-set theory postulates that all things have something in common with at
least one other thing. The difficulty in quantifying these sets gives rise to the
name "rough sets." These sets provide a way of comparing and drawing
conclusions regarding operational elements that may not be fully defined or
quantified. An example of rough sets used in operations is a historical comparison
of the Great Depression of the 1930s with the Credit Crisis of 2008. These sets are
not the same, but they do intersect at certain points, and these intersections can
reveal important clues to the effect of management operations before and during
periods of financial crises
A study published in the "International Journal of Mathematical Models and
Methods in Applied Sciences" used rough-set theory to study historical bank
failures. The study was conducted to assist in predicting a potential failure of
Turkish banks. Since every economy depends on the health of its banks, rough-set
theory is useful. Using rough-set theory, banks can form an early warning system
of potential bank failures, improving financial management.