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DISCRETE COMPUTATIONAL
STRUCTURES
CSE 2353
Fall 2011
Most slides modified from
Discrete Mathematical Structures: Theory and Applications
CSE 2353 OUTLINE
1.
2.
3.
4.
5.
6.
7.
8.
Sets
Logic
Proof Techniques
Integers and Induction
Relations and Posets
Functions
Counting Principles
Boolean Algebra
CSE 2353 OUTLINE
1.Sets
2.
3.
4.
5.
6.
7.
8.
Logic
Proof Techniques
Integers and Induction
Relations and Posets
Functions
Counting Principles
Boolean Algebra
Sets: Learning Objectives
Learn about sets
Explore various operations on sets
Become familiar with Venn diagrams
CS:
Learn how to represent sets in computer memory
Learn how to implement set operations in programs
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Sets
Definition: Well-defined collection of distinct
objects
Members or Elements: part of the collection
Roster Method: Description of a set by listing the
elements, enclosed with braces
Examples:
Vowels = {a,e,i,o,u}
Primary colors = {red, blue, yellow}
Membership examples
“a belongs to the set of Vowels” is written as: a 
Vowels
“j does not belong to the set of Vowels: j  Vowels
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Sets
Set-builder method
A = { x | x  S, P(x) } or A = { x  S | P(x) }
 A is the set of all elements x of S, such that x
satisfies the property P
Example:
If X = {2,4,6,8,10}, then in set-builder notation, X
can be described as
X = {n  Z | n is even and 2  n  10}
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Sets
 Standard Symbols which denote sets of numbers
 N : The set of all natural numbers (i.e.,all positive integers)
 Z : The set of all integers
 Z+ : The set of all positive integers
 Z* : The set of all nonzero integers
 E : The set of all even integers
 Q : The set of all rational numbers
 Q* : The set of all nonzero rational numbers
 Q+ : The set of all positive rational numbers
 R : The set of all real numbers
 R* : The set of all nonzero real numbers
 R+ : The set of all positive real numbers
 C : The set of all complex numbers
 C* : The set of all nonzero complex numbers
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Sets
Subsets
“X is a subset of Y” is written as X  Y
“X is not a subset of Y” is written as X
Y
Example:
 X = {a,e,i,o,u}, Y = {a, i, u} and z = {b,c,d,f,g}
Y  X, since every element of Y is an element of X
Y
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Z, since a  Y, but a  Z
Sets
Superset
X and Y are sets. If X  Y, then “X is contained in
Y” or “Y contains X” or Y is a superset of X,
written Y  X
Proper Subset
X and Y are sets. X is a proper subset of Y if X 
Y and there exists at least one element in Y that is
not in X. This is written X  Y.
Example:
 X = {a,e,i,o,u}, Y = {a,e,i,o,u,y}
X  Y , since y  Y, but y  X
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Sets
Set Equality
X and Y are sets. They are said to be equal if every
element of X is an element of Y and every element
of Y is an element of X, i.e. X  Y and Y  X
Examples:
{1,2,3} = {2,3,1}
X = {red, blue, yellow} and Y = {c | c is a primary
color} Therefore, X=Y
Empty (Null) Set
A Set is Empty (Null) if it contains no elements.
The Empty Set is written as 
The Empty Set is a subset of every set
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Sets
Finite and Infinite Sets
X is a set. If there exists a nonnegative integer n
such that X has n elements, then X is called a
finite set with n elements.
If a set is not finite, then it is an infinite set.
Examples:
 Y = {1,2,3} is a finite set
 P = {red, blue, yellow} is a finite set
 E , the set of all even integers, is an infinite set
  , the Empty Set, is a finite set with 0 elements
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Sets
Cardinality of Sets
Let S be a finite set with n distinct elements,
where n ≥ 0. Then |S| = n , where the cardinality
(number of elements) of S is n
Example:
If P = {red, blue, yellow}, then |P| = 3
Singleton
 A set with only one element is a singleton
Example:
H = { 4 }, |H| = 1, H is a singleton
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Sets
Power Set
For any set X ,the power set of X ,written P(X),is
the set of all subsets of X
Example:
If X = {red, blue, yellow}, then P(X) = {  , {red},
{blue}, {yellow}, {red,blue}, {red, yellow}, {blue,
yellow}, {red, blue, yellow} }
Universal Set
An arbitrarily chosen, but fixed set
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Sets
Venn Diagrams
Abstract visualization
of a Universal set, U
as a rectangle, with all
subsets of U shown as
circles.
Shaded portion
represents the
corresponding set
Example:
In Figure 1, Set X,
shaded, is a subset of
the Universal set, U
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Set Operations and Venn Diagrams
Union of Sets
Example:
If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then
XUY = {1,2,3,4,5,6,7,8,9}
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Sets
Intersection of Sets
Example:
If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then X ∩ Y = {5}
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Sets
Disjoint Sets
Example:
If X = {1,2,3,4,} and Y = {6,7,8,9}, then X ∩ Y = 
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Sets
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Sets
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Sets
Difference
• Example:
If X = {a,b,c,d} and Y =
{c,d,e,f}, then X – Y =
{a,b} and Y – X = {e,f}
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Sets
Complement
Example:
If U = {a,b,c,d,e,f} and X = {c,d,e,f}, then X’ = {a,b}
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Sets
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Sets
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Sets
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Sets
Ordered Pair
X and Y are sets. If x  X and y  Y, then an ordered
pair is written (x,y)
Order of elements is important. (x,y) is not necessarily
equal to (y,x)
Cartesian Product
 The Cartesian product of two sets X and Y ,written X × Y
,is the set
 X × Y ={(x,y)|x ∈ X , y ∈ Y}
For any set X, X ×  =  =  × X
 Example:
 X = {a,b}, Y = {c,d}
X × Y = {(a,c), (a,d), (b,c), (b,d)}
Y × X = {(c,a), (d,a), (c,b), (d,b)}
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Computer Representation of Sets
 A Set may be stored in a computer in an array as an
unordered list
Problem: Difficult to perform operations on the set.
Linked List
 Solution: use Bit Strings (Bit Map)
A Bit String is a sequence of 0s and 1s
Length of a Bit String is the number of digits in the
string
Elements appear in order in the bit string
A 0 indicates an element is absent, a 1 indicates that the
element is present
 A set may be implemented as a file
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Computer Implementation of Set Operations
Bit Map
File
Operations
Intersection
Union
Element of
Difference
Complement
Power Set
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Special “Sets” in CS
Multiset
Ordered Set
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CSE 2353 OUTLINE
1. Sets
2.Logic
3. Proof Techniques
4. Relations and Posets
5. Functions
6. Counting Principles
7. Boolean Algebra
Logic: Learning Objectives
 Learn about statements (propositions)
 Learn how to use logical connectives to combine statements
 Explore how to draw conclusions using various argument forms
 Become familiar with quantifiers and predicates
 CS
 Boolean data type
 If statement
 Impact of negations
 Implementation of quantifiers
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Mathematical Logic
Definition: Methods of reasoning, provides rules
and techniques to determine whether an
argument is valid
Theorem: a statement that can be shown to be
true (under certain conditions)
Example: If x is an even integer, then x + 1 is an
odd integer
This statement is true under the condition that x is
an integer is true
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Mathematical Logic
A statement, or a proposition, is a declarative
sentence that is either true or false, but not both
Lowercase letters denote propositions
Examples:
p: 2 is an even number (true)
q: 3 is an odd number (true)
r: A is a consonant (false)
The following are not propositions:
p: My cat is beautiful
q: Are you in charge?
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Mathematical Logic
 Truth value
 One of the values “truth” (T) or “falsity” (F) assigned to a
statement
 Negation
 The negation of p, written ~p, is the statement obtained
by negating statement p
Example:
p: A is a consonant
~p: it is the case that A is not a consonant
 Truth Table
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Mathematical Logic
Conjunction
Let p and q be statements.The conjunction of p
and q, written p ^ q , is the statement formed by
joining statements p and q using the word “and”
The statement p ^ q is true if both p and q are
true; otherwise p ^ q is false
Truth Table for
Conjunction:
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Mathematical Logic
Disjunction
Let p and q be statements. The disjunction of p
and q, written p v q , is the statement formed by
joining statements p and q using the word “or”
The statement p v q is true if at least one of the
statements p and q is true; otherwise p v q is
false
The symbol v is read “or”
Truth Table for Disjunction:
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Mathematical Logic
 Implication
Let p and q be statements.The statement “if p then
q” is called an implication or condition.
The implication “if p then q” is written p  q
 “If p, then q””
p is called the hypothesis, q is called the
conclusion
 Truth Table for
Implication:
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Mathematical Logic
Implication
Let p: Today is Sunday and q: I will wash the car.
p  q :
If today is Sunday, then I will wash the car
The converse of this implication is written q  p
If I wash the car, then today is Sunday
The inverse of this implication is ~p  ~q
If today is not Sunday, then I will not wash the car
The contrapositive of this implication is ~q  ~p
If I do not wash the car, then today is not Sunday
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Mathematical Logic
Biimplication
Let p and q be statements. The statement “p if and
only if q” is called the biimplication or
biconditional of p and q
The biconditional “p if and only if q” is written p  q
“p if and only if q”
Truth Table for the
Biconditional:
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Mathematical Logic
 Statement Formulas
 Definitions
 Symbols p ,q ,r ,...,called statement variables
 Symbols ~, , v, →,and ↔ are called logical
^
connectives
1) A statement variable is a statement formula
2) If A and B are statement formulas, then the
expressions (~A ), (A B) , (A v B ), (A → B )
^
and (A ↔ B ) are statement formulas
 Expressions are statement formulas that are
constructed only by using 1) and 2) above
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Mathematical Logic
Precedence of logical connectives is:
~ highest

^
second highest
 v third highest
→ fourth highest
↔ fifth highest
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Mathematical Logic
Tautology
A statement formula A is said to be a tautology if
the truth value of A is T for any assignment of the
truth values T and F to the statement variables
occurring in A
Contradiction
A statement formula A is said to be a
contradiction if the truth value of A is F for any
assignment of the truth values T and F to the
statement variables occurring in A
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Mathematical Logic
Logically Implies
A statement formula A is said to logically imply a
statement formula B if the statement formula A →
B is a tautology. If A logically implies B, then
symbolically we write A → B
Logically Equivalent
A statement formula A is said to be logically
equivalent to a statement formula B if the
statement formula A ↔ B is a tautology. If A is
logically equivalent to B , then symbolically we
write A ≡ B
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Mathematical Logic
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Validity of Arguments
Proof: an argument or a proof of a theorem
consists of a finite sequence of statements
ending in a conclusion
Argument: a finite sequence A1 , A2 , A3 , ..., An1 , An
of statements.
The final statement, An , is the conclusion, and
the statements A1 , A2 , A3 , ..., An 1 are the
premises of the argument.
An argument is logically valid if the statement
formula A1 , A2 , A3 , ..., An1  An is a tautology.
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Validity of Arguments
Valid Argument Forms
Modus Ponens:
Modus Tollens :
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Validity of Arguments
Valid Argument Forms
Disjunctive Syllogisms:
Hypothetical Syllogism:
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Validity of Arguments
Valid Argument Forms
Dilemma:
Conjunctive Simplification:
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Validity of Arguments
Valid Argument Forms
Disjunctive Addition:
Conjunctive Addition:
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Quantifiers and First Order Logic
Predicate or Propositional Function
Let x be a variable and D be a set; P(x) is a
sentence
Then P(x) is called a predicate or propositional
function with respect to the set D if for each
value of x in D, P(x) is a statement; i.e., P(x) is
true or false
Moreover, D is called the domain of the
discourse and x is called the free variable
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Quantifiers and First Order Logic
Universal Quantifier
Let P(x) be a predicate and let D be the domain
of the discourse. The universal quantification of
P(x) is the statement:
For all x, P(x)
or
For every x, P(x)
The symbol  is read as “for all and every”
 x P ( x )
 Two-place predicate: xy P( x, y )
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Quantifiers and First Order Logic
Existential Quantifier
Let P(x) be a predicate and let D be the domain
of the discourse. The existential quantification of
P(x) is the statement:
There exists x, P(x)
The symbol  is read as “there exists”
 x P ( x )
 Bound Variable
The variable appearing in:
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x P ( x)
or
x P ( x )
Quantifiers and First Order Logic
Negation of Predicates (DeMorgan’s Laws)

~ x P( x)  x ~ P( x)
Example:
 If P(x) is the statement “x has won a race” where
the domain of discourse is all runners, then the
universal quantification of P(x) is x P ( x ) , i.e.,
every runner has won a race. The negation of this
statement is “it is not the case that every runner
has won a race. Therefore there exists at least one
runner who has not won a race. Therefore: x ~ P ( x)
and so,
~ x P( x)  x ~ P( x)
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Quantifiers and First Order Logic
Negation of Predicates (DeMorgan’s
Laws)
 ~ x P( x)  x ~ P( x)
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Arguments in Predicate Logic
Universal Specification
If x F ( x) is true, then a  U F(a) is true
Universal Generalization
If F(a) is true a  U then x F ( x) is true
Existential Specification
If x F ( x ) is true, then a  U where F(a) is true
Existential Generalization
If F(a) is true a  U then x F ( x ) is true
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54
Logic and CS
Logic is basis of ALU
Logic is crucial to IF statements
AND
OR
NOT
Implementation of quantifiers
Looping
Database Query Languages
Relational Algebra
Relational Calculus
SQL
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CSE 2353 OUTLINE
1. Sets
2. Logic
3. Proof Techniques
4. Integers and Inductions
5. Relations and Posets
6. Functions
7. Counting Principles
8. Boolean Algebra
Proof Technique: Learning Objectives
 Learn various proof techniques
 Direct
 Indirect
 Contradiction
 Induction
 Practice writing proofs
CS: Why study proof techniques?
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Proof Techniques
Theorem
Statement that can be shown to be true (under
certain conditions)
Typically Stated in one of three ways
As Facts
As Implications
As Biimplications
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Proof Techniques
Direct Proof or Proof by Direct Method
Proof of those theorems that can be expressed in
the form ∀x (P(x) → Q(x)), D is the domain of
discourse
Select a particular, but arbitrarily chosen, member
a of the domain D
Show that the statement P(a) → Q(a) is true.
(Assume that P(a) is true
Show that Q(a) is true
By the rule of Universal Generalization (UG),
∀x (P(x) → Q(x)) is true
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Proof Techniques
Indirect Proof
The implication p → q is equivalent to the
implication (∼q → ∼p)
Therefore, in order to show that p → q is true,
one can also show that the implication
(∼q → ∼p) is true
To show that (∼q → ∼p) is true, assume that the
negation of q is true and prove that the negation
of p is true
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Proof Techniques
Proof by Contradiction
Assume that the conclusion is not true and then
arrive at a contradiction
Example: Prove that there are infinitely many prime
numbers
Proof:
Assume there are not infinitely many prime numbers,
therefore they are listable, i.e. p1,p2,…,pn
Consider the number q = p1p2…pn+1. q is not
divisible by any of the listed primes
Therefore, q is a prime. However, it was not listed.
Contradiction! Therefore, there are infinitely many
primes.
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Proof Techniques
Proof of Biimplications
To prove a theorem of the form ∀x (P(x) ↔
Q(x )), where D is the domain of the
discourse, consider an arbitrary but fixed
element a from D. For this a, prove that the
biimplication P(a) ↔ Q(a) is true
The biimplication p ↔ q is equivalent to
(p → q) ∧ (q → p)
Prove that the implications p → q and q → p
are true
Assume that p is true and show that q is true
Assume that q is true and show that p is true
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Proof Techniques
Proof of Equivalent Statements
Consider the theorem that says that
statements p,q and r are equivalent
Show that p → q, q → r and r → p
Assume p and prove q. Then assume q
and prove r Finally, assume r and prove p
What other methods are possible?
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Other Proof Techniques
Vacuous
Trivial
Contrapositive
Counter Example
Divide into Cases
Constructive
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Proof Basics
You can not prove by
example
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Proofs in Computer Science
Proof of program correctness
Proofs are used to verify approaches
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CSE 2353 OUTLINE
1. Sets
2. Logic
3. Proof Techniques
4. Integers and Induction
5. Relations and Posets
6. Functions
7. Counting Principles
8. Boolean Algebra
Learning Objectives
Learn how the principle of mathematical
induction is used to solve problems and proofs
Learn about the basic properties of integers
Explore how addition and subtraction operations
are performed on binary numbers
CS
Become aware how integers are represented in
computer memory
Looping
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Mathematical Deduction
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Mathematical Deduction
 Proof of a mathematical statement by the principle of
mathematical induction consists of three steps:
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Mathematical Deduction
Assume that when a domino is knocked over, the
next domino is knocked over by it
Show that if the first domino is knocked over,
then all the dominoes will be knocked over
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Mathematical Deduction
Let P(n) denote the statement that then nth
domino is knocked over
Show that P(1) is true
Assume some P(k) is true, i.e. the kth domino is
knocked over for some
k 1
Prove that P(k+1) is true, i.e.
P ( k )  P ( k  1)
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Mathematical Deduction
Assume that when a staircase is climbed, the
next staircase is also climbed
Show that if the first staircase is climbed then
all staircases can be climbed
Let P(n) denote the statement that then nth
staircase is climbed
It is given that the first staircase is climbed, so
P(1) is true
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Mathematical Deduction
Suppose some P(k) is true, i.e. the kth
staircase is climbed for some k  1
By the assumption, because the kth staircase
was climbed, the k+1st staircase was
climbed
Therefore, P(k) is true, so
P ( k )  P ( k  1)
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Mathematical Deduction
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Mathematical Deduction
We can associate a predicate, P(n). The
predicate P(n) is such that:
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Integers
Properties of Integers
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Integers
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Integers
The div and mod operators
div
 a div b = the quotient of a and b obtained by dividing a
on b.
Examples:
8 div 5 = 1
13 div 3 = 4
mod
a mod b = the remainder of a and b obtained by dividing
a on b
8 mod 5 = 3
13 mod 3 = 1
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Integers
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Integers
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Integers
 Relatively Prime
Number
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Integers
 Least Common Multiples
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Representation of Integers in Computers
Digital Signals
0s and 1s – 0s represent low voltage, 1s high
voltage
Digital signals are more reliable carriers of
information than analog signals
Can be copied from one device to another
with exact precision
Machine language is a sequence of 0s and 1s
The digit 0 or 1 is called a binary digit , or bit
A sequence of 0s and 1s is sometimes referred to
as binary code
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Representation of Integers in Computers
Decimal System or Base-10
The digits that are used to represent numbers in base
10 are 0,1,2,3,4,5,6,7,8, and 9
Binary System or Base-2
Computer memory stores numbers in machine
language, i.e., as a sequence of 0s and 1s
Octal System or Base-8
Digits that are used to represent numbers in base 8
are 0,1,2,3,4,5,6, and 7
Hexadecimal System or Base-16
Digits and letters that are used to represent numbers
in base 16 are 0,1,2,3,4,5,6,7,8,9,A ,B ,C ,D ,E , and
F
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Representation of Integers in Computers
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Representation of Integers in Computers
Two’s Complements and Operations
on Binary Numbers
In computer memory, integers are
represented as binary numbers in fixedlength bit strings, such as 8, 16, 32 and 64
Assume that integers are represented as
8-bit fixed-length strings
Sign bit is the MSB (Most Significant Bit)
Leftmost bit (MSB) = 0, number is positive
Leftmost bit (MSB) = 1, number is negative
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Representation of Integers in Computers
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Representation of Integers in Computers
One’s Complements and Operations on Binary
Numbers
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Representation of Integers in Computers
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Prime Numbers
Example:
Consider the integer 131. Observe that 2 does not divide 131. We now find all
odd primes p such that p2  131. These primes are 3, 5, 7, and 11. Now none of
3, 5, 7, and 11 divides 131. Hence, 131 is a prime.
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Prime Numbers
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Prime Numbers
Factoring a Positive Integer
The standard factorization of n
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Prime Numbers
 Fermat’s Factoring Method
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Prime Numbers
 Fermat’s Factoring Method
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CSE 2353 OUTLINE
1. Sets
2. Logic
3. Proof Techniques
4. Integers and Induction
5.Relations and Posets
6. Functions
7. Counting Principles
8. Boolean Algebra
Learning Objectives
Learn about relations and their basic
properties
Explore equivalence relations
Become aware of closures
Learn about posets
Explore how relations are used in the design
of relational databases
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Relations
Relations are a natural way to associate
objects of various sets
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Relations
 R can be described in
 Roster form
 Set-builder form
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Relations
Arrow Diagram
Write the elements of A in one column
Write the elements B in another column
Draw an arrow from an element, a, of A to an element,
b, of B, if (a ,b)  R
Here, A = {2,3,5} and B = {7,10,12,30} and R from A
into B is defined as follows: For all a  A and b  B,
a R b if and only if a divides b
The symbol → (called an arrow) represents the relation
R
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Relations
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Relations
Directed Graph
Let R be a relation on a finite set A
Describe R pictorially as follows:
For each element of A , draw a small or big
dot and label the dot by the corresponding
element of A
Draw an arrow from a dot labeled a , to
another dot labeled, b , if a R b .
Resulting pictorial representation of R is called
the directed graph representation of the
relation R
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Relations
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Relations
Domain and Range of the Relation
Let R be a relation from a set A into a set B.
Then R ⊆ A x B. The elements of the relation R
tell which element of A is R-related to which
element of B
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Relations
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Relations
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Relations
Let A = {1, 2, 3, 4} and B = {p, q, r}. Let R = {(1, q), (2, r
), (3, q), (4, p)}. Then R−1 = {(q, 1), (r , 2), (q, 3), (p,
4)}
To find R−1, just reverse the directions of the arrows
D(R) = {1, 2, 3, 4} = Im(R−1), Im(R) = {p, q, r} = D(R−1)
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Relations
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Relations
Constructing New Relations from
Existing Relations
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Relations
Example:
Consider the relations R and S as given in
Figure 3.7.
The composition S ◦ R is given by Figure 3.8.
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Relations
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
Hasse Diagram
Let S = {1, 2, 3}. Then P(S)
= {, {1}, {2}, {3}, {1, 2}, {2,
3}, {1, 3}, S}
Now (P(S),≤) is a poset,
where ≤ denotes the set
inclusion relation. The
poset diagram of (P(S),≤) is
shown in Figure 3.22
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Partially Ordered Sets
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Partially Ordered Sets
Hasse Diagram
Let S = {1, 2, 3}. Then
P(S) = {, {1}, {2}, {3}, {1,
2}, {2, 3}, {1, 3}, S}
(P(S),≤) is a poset, where
≤ denotes the set
inclusion relation
Draw the digraph of this
inclusion relation (see
Figure 3.23). Place the
vertex A above vertex B if
B ⊂ A. Now follow steps
(2), (3), and (4)
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Partially Ordered Sets
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Partially Ordered Sets
Hasse Diagram
Consider the poset (S,≤), where
S = {2, 4, 5, 10, 15, 20} and the
partial order ≤ is the divisibility
relation.
2 and 5 are the only minimal
elements of this poset.
This poset has no least element.
20 and 15 are the only maximal
elements of this poset.
This poset has no greatest
element.
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Partially Ordered Sets
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Application: Relational Database
A database is a shared and integrated
computer structure that stores
End-user data; i.e., raw facts that are of interest
to the end user;
Metadata, i.e., data about data through which
data are integrated
A database can be thought of as a well-organized
electronic file cabinet whose contents are
managed by software known as a database
management system; that is, a collection of
programs to manage the data and control the
accessibility of the data
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Application: Relational Database
In a relational database system,
tables are considered as relations
A table is an n-ary relation, where n is
the number of columns in the tables
The headings of the columns of a table
are called attributes, or fields, and
each row is called a record
The domain of a field is the set of all
(possible) elements in that column
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Application: Relational Database
Each entry in the ID column uniquely
identifies the row containing that ID
Such a field is called a primary key
Sometimes, a primary key may consist of
more than one field
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Application: Relational Database
Structured Query Language (SQL)
Information from a database is retrieved via a
query, which is a request to the database for
some information
A relational database management system
provides a standard language, called
structured query language (SQL)
A relational database management system
provides a standard language, called
structured query language (SQL)
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Application: Relational Database
Structured Query Language (SQL)
An SQL contains commands to create tables, insert
data into tables, update tables, delete tables, etc.
Once the tables are created, commands can be used
to manipulate data into those tables.
The most commonly used command for this purpose
is the select command. The select command allows
the user to do the following:
Specify what information is to be retrieved and from which
tables.
Specify conditions to retrieve the data in a specific form.
Specify how the retrieved data are to be displayed.
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CSE 2353 OUTLINE
1. Sets
2. Logic
3. Proof Techniques
4. Integers and Induction
5. Relations and Posets
6.Functions
7. Counting Principles
8. Boolean Algebra
Learning Objectives
Learn about functions
Explore various properties of functions
Learn about binary operations
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Functions
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Functions
Every function is a relation
Therefore, functions on
finite sets can be described
by arrow diagrams. In the
case of functions, the
arrow diagram may be
drawn slightly differently.
If f : A → B is a function
from a finite set A into a
finite set B, then in the
arrow diagram, the
elements of A are enclosed
in ellipses rather than
individual boxes.
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Functions
 To determine from its arrow diagram
whether a relation f from a set A into
a set B is a function, two things are
checked:
1) Check to see if there is an arrow from
each element of A to an element of B
 This would ensure that the domain of f is the
set A, i.e., D(f) = A
2) Check to see that there is only one
arrow from each element of A to an
element of B
 This would ensure that f is well defined
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Functions
Let A = {1,2,3,4} and B =
{a, b, c , d} be sets
The arrow diagram in
Figure 5.6 represents the
relation f from A into B
Every element of A has
some image in B
An element of A is
related to only one
element of B; i.e., for
each a ∈ A there exists a
unique element b ∈ B
such that f (a) = b
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Functions
Therefore, f is a function
from A into B
The image of f is the set
Im(f) = {a, b, d}
There is an arrow
originating from each
element of A to an
element of B
 D(f) = A
There is only one arrow
from each element of A
to an element of B
 f is well defined
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Functions
The arrow diagram in
Figure 5.7 represents the
relation g from A into B
Every element of A has
some image in B
 D(g ) = A
For each a ∈ A, there
exists a unique element b
∈ B such that g(a) = b
 g is a function from A into
B
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Functions
The image of g is
Im(g) = {a, b, c , d}
=B
There is only one
arrow from each
element of A to an
element of B
 g is well defined
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Functions
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Functions
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Functions
Example 5.1.16
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 Let A = {1,2,3,4} and B = {a, b, c ,
d}. Let f : A → B be a function such
that the arrow diagram of f is as
shown in Figure 5.10
 The arrows from a distinct element
of A go to a distinct element of B.
That is, every element of B has at
most one arrow coming to it.
 If a1, a2 ∈ A and a1 = a2, then
f(a1) = f(a2). Hence, f is oneone.
 Each element of B has an arrow
coming to it. That is, each element
of B has a preimage.
 Im(f) = B. Hence, f is onto B. It
also follows that f is a one-toone correspondence.
Functions
Example 5.1.18
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Let A = {1,2,3,4} and
B = {a , b , c , d , e }
 f : 1 → a, 2 → a, 3 → a,
4 →a
For this function the
images of distinct
elements of the domain
are not distinct. For
example 1  2, but f(1)
= a = f(2) .
Im(f) = {a}  B. Hence, f
is neither one-one nor
onto B.
Functions
Let A = {1,2,3,4} and
B = {a, b, c , d, e }
 f : 1 → a, 2 → b, 3 → d,
4→e
 f is one-one. In this
function, for the element
c of B, the codomain,
there is no element x in
the domain such that f(x)
= c ; i.e., c has no
preimage. Hence, f is
not onto B.
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Functions
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Functions
Let A = {1,2,3,4}, B = {a, b, c , d, e},and C =
{7,8,9}. Consider the functions f : A → B, g :
B → C as defined by the arrow diagrams in
Figure 5.14.
The arrow diagram in Figure 5.15 describes
the function
h = g ◦ f : A → C.
CSE 2353 f11
Special Functions and Cardinality of a
Set
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Special Functions and Cardinality of a Set
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Special Functions and Cardinality of a
Set
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Special Functions and Cardinality of a
Set
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Special Functions and Cardinality of a
Set
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Special Functions and Cardinality of a
Set
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Special Functions and Cardinality of a
Set
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Special Functions and Cardinality of a
Set
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Special Functions and Cardinality of a
Set
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Special Functions and Cardinality of a
Set
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Special Functions and Cardinality of a
Set
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Binary Operations
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CSE 2353 OUTLINE
1. Sets
2. Logic
3. Proof Techniques
4. Integers and Induction
5. Relations and Posets
6. Functions
7.Counting Principles
8. Boolean Algebra
Learning Objectives
Learn the basic counting principles—
multiplication and addition
Explore the pigeonhole principle
Learn about permutations
Learn about combinations
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Basic Counting Principles
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Basic Counting Principles
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Pigeonhole Principle
The pigeonhole principle is also known as the
Dirichlet drawer principle, or the shoebox
principle.
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Pigeonhole Principle
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Permutations
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Permutations
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Combinations
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Combinations
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Generalized Permutations and Combinations
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CSE 2353 OUTLINE
1. Sets
2. Logic
3. Proof Techniques
4. Integers and Induction
5. Relations and Posets
6. Functions
7. Counting Principles
8.Boolean Algebra
Two-Element Boolean Algebra
Let B = {0, 1}.
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Two-Element Boolean Algebra
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Two-Element Boolean Algebra
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Boolean Algebra
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Boolean Algebra
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
The Karnaugh map, or K-map for short, can be
used to minimize a sum-of-product Boolean
expression.
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