1. Foundations: Sets, Logic, and Algorithms
Discrete Mathematical Structures:
Theory and Applications

2. Learning Objectives Learn about sets
Explore various operations on sets
Become familiar with Venn diagrams
Learn how to represent sets in computer memory
Learn about statements (propositions)
Discrete Mathematical Structures: Theory and Applications

3. Learning Objectives
Learn how to use logical connectives to combine statements
Explore how to draw conclusions using various argument forms
Become familiar with quantifiers and predicates
Learn various proof techniques
Explore what an algorithm is
Discrete Mathematical Structures: Theory and Applications

4. 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
Discrete Mathematical Structures: Theory and Applications

5. 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}
Discrete Mathematical Structures: Theory and Applications

6. 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 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
Discrete Mathematical Structures: Theory and Applications

7. 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 Z, since a  Y, but a  Z
Discrete Mathematical Structures: Theory and Applications

8. Sets Superset Proper Subset
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
Discrete Mathematical Structures: Theory and Applications

9. Sets Set Equality Empty (Null) Set
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
Discrete Mathematical Structures: Theory and Applications

10. 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
Discrete Mathematical Structures: Theory and Applications

11. 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
H = { 4 }, |H| = 1, H is a singleton
Discrete Mathematical Structures: Theory and Applications

12. Sets Power Set Universal 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
Discrete Mathematical Structures: Theory and Applications

13. 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
Discrete Mathematical Structures: Theory and Applications

14. Sets Union of Sets Example:
If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then
X∪Y = {1,2,3,4,5,6,7,8,9}
Discrete Mathematical Structures: Theory and Applications

15. Sets Intersection of Sets Example:
If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then X ∩ Y = {5}
Discrete Mathematical Structures: Theory and Applications

16. Sets Disjoint Sets Example:
If X = {1,2,3,4,} and Y = {6,7,8,9}, then X ∩ Y = 
Discrete Mathematical Structures: Theory and Applications

17. Sets
Discrete Mathematical Structures: Theory and Applications

18. Sets
Discrete Mathematical Structures: Theory and Applications

19. Sets
The union and intersection of three,four,or even infinitely many sets can be considered
For a finite collection of n sets, X1, X2, … Xn where n ≥ 2 :
Discrete Mathematical Structures: Theory and Applications

20. Sets
Index Set
Discrete Mathematical Structures: Theory and Applications

21. Sets
Example:
If A = {a,b,c}, B = {x, y, z} and C = {1,2,3} then A ∩ B =  and B ∩ C =  and A ∩ C = . Therefore, A,B,C are pairwise disjoint
Discrete Mathematical Structures: Theory and Applications

22. 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}
Discrete Mathematical Structures: Theory and Applications

23. Sets Complement Example:
If U = {a,b,c,d,e,f} and X = {c,d,e,f}, then X’ = {a,b}
Discrete Mathematical Structures: Theory and Applications

24. Sets
Discrete Mathematical Structures: Theory and Applications

25. Sets
Discrete Mathematical Structures: Theory and Applications

26. Sets
Discrete Mathematical Structures: Theory and Applications

27. Sets Ordered Pair Cartesian Product
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)}
Discrete Mathematical Structures: Theory and Applications

28. Sets Diagonal of a Set δx = {(x,x) | x ∈ X}
For a set X ,the set δx , is the diagonal of X, defined by
δx = {(x,x) | x ∈ X}
Example:
X = {a,b,c}, δx = {(a,a), (b,b), (c,c)}
Discrete Mathematical Structures: Theory and Applications

29. Sets 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.
Solution: use Bit Strings
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
Discrete Mathematical Structures: Theory and Applications

30. 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
Discrete Mathematical Structures: Theory and Applications

31. 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?
Discrete Mathematical Structures: Theory and Applications

32. Mathematical Logic Truth value Negation
One of the values “truth” or “falsity” assigned to a statement
True is abbreviated to T or 1
False is abbreviated to F or 0
Negation
The negation of p, written ∼p, is the statement obtained by negating statement p
Truth values of p and ∼p are opposite
Symbol ~ is called “not” ~p is read as as “not p”
Example:
p: A is a consonant
~p: it is the case that A is not a consonant
q: Are you in charge?
Discrete Mathematical Structures: Theory and Applications

33. Mathematical Logic Truth Table 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
Discrete Mathematical Structures: Theory and Applications

34. Mathematical Logic Conjunction Truth Table for Conjunction:
Discrete Mathematical Structures: Theory and Applications

35. Mathematical Logic Disjunction
Let p and q be statements. The disjunction of p and q, written p ∨ q , is the statement formed by joining statements p and q using the word “or”
The statement p∨q is true if at least one of the statements p and q is true; otherwise p∨q is false
The symbol ∨ is read “or”
Discrete Mathematical Structures: Theory and Applications

36. Mathematical Logic Disjunction Truth Table for Disjunction:
Discrete Mathematical Structures: Theory and Applications

37. 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
p  q is read:
“If p, then q”
“p is sufficient for q”
q if p
q whenever p
Discrete Mathematical Structures: Theory and Applications

38. Mathematical Logic Implication Truth Table for Implication:
p is called the hypothesis, q is called the conclusion
Discrete Mathematical Structures: Theory and Applications

39. Mathematical Logic Implication
Let p: Today is Sunday and q: I will wash the car. The conjunction p  q is the statement:
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
Discrete Mathematical Structures: Theory and Applications

40. 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  q is read:
“p if and only if q”
“p is necessary and sufficient for q”
“q if and only if p”
“q when and only when p”
Discrete Mathematical Structures: Theory and Applications

41. Mathematical Logic Biconditional Truth Table for the Biconditional:
Discrete Mathematical Structures: Theory and Applications

42. Mathematical Logic Statement Formulas Definitions
Symbols p ,q ,r ,...,called statement variables
Symbols ~, ∧, ∨, →,and ↔ are called logical connectives
A statement variable is a statement formula
If A and B are statement formulas, then the expressions (~A ), (A ∧ B) , (A ∨ B ), (A → B ) and (A ↔ B ) are statement formulas
Expressions are statement formulas that are constructed only by using 1) and 2) above
Discrete Mathematical Structures: Theory and Applications

43. Mathematical Logic Precedence of logical connectives is:
~ highest
∧ second highest
∨ third highest
→ fourth highest
↔ fifth highest
Discrete Mathematical Structures: Theory and Applications

44. Mathematical Logic Example:
Let A be the statement formula (~(p ∨q )) → (q ∧p )
Truth Table for A is:
Discrete Mathematical Structures: Theory and Applications

45. Mathematical Logic Tautology Contradiction
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
Discrete Mathematical Structures: Theory and Applications

46. Mathematical Logic Logically Implies Logically Equivalent
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 (or A ⇔ B)
Discrete Mathematical Structures: Theory and Applications

47. Mathematical Logic
Discrete Mathematical Structures: Theory and Applications

48. Mathematical Logic Proof of (~p ∧q ) → (~(q →p ))
Discrete Mathematical Structures: Theory and Applications

49. Mathematical Logic Proof of (~p ∧q ) → (~(q →p )) [Continued]
Discrete Mathematical Structures: Theory and Applications

50. 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
of statements.
The final statement, , is the conclusion, and the statements are the premises of the argument.
An argument is logically valid if the statement formula
is a tautology.
Discrete Mathematical Structures: Theory and Applications

51. Validity of Arguments Valid Argument Forms
Modus Ponens (Method of Affirming)
Modus Tollens (Method of Denying)
Discrete Mathematical Structures: Theory and Applications

52. Validity of Arguments Valid Argument Forms Disjunctive Syllogisms
Discrete Mathematical Structures: Theory and Applications

53. Validity of Arguments Valid Argument Forms Hypothetical Syllogism
Dilemma
Discrete Mathematical Structures: Theory and Applications

54. Validity of Arguments Valid Argument Forms Conjunctive Simplification
Discrete Mathematical Structures: Theory and Applications

55. Validity of Arguments Valid Argument Forms Disjunctive Addition
Discrete Mathematical Structures: Theory and Applications

56. Validity of Arguments Valid Argument Forms Conjunctive Addition
Discrete Mathematical Structures: Theory and Applications

57. 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
Discrete Mathematical Structures: Theory and Applications

58. Quantifiers and First Order Logic
Predicate or Propositional Function
Example:
Q(x,y) : x > y, where the Domain is the set of integers
Q is a 2-place predicate
Q is T for Q(4,3) and Q is F for Q (3,4)
Discrete Mathematical Structures: Theory and Applications

59. 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”
Two-place predicate:
Discrete Mathematical Structures: Theory and Applications

60. 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”
Bound Variable
The variable appearing in: or
Discrete Mathematical Structures: Theory and Applications

61. Quantifiers and First Order Logic
Negation of Predicates (DeMorgan’s Laws)
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 , 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:
and so,
Discrete Mathematical Structures: Theory and Applications

62. Quantifiers and First Order Logic
Negation of Predicates (DeMorgan’s Laws)
Discrete Mathematical Structures: Theory and Applications

63. Quantifiers and First Order Logic
Formulas in Predicate Logic
All statement formulas are considered formulas
Each n, n =1,2,...,n-place predicate P( ) containing the variables is a formula.
If A and B are formulas, then the expressions ~A, (A∧B), (A∨B) , A →B and A↔B are statement formulas, where ~, ∧, ∨, → and ↔ are logical connectives
If A is a formula and x is a variable, then ∀x A(x) and ∃x A(x) are formulas
All formulas constructed using only above rules are considered formulas in predicate logic
Discrete Mathematical Structures: Theory and Applications

64. Quantifiers and First Order Logic
Additional Rules of Inference
If the statement ∀x P(x) is assumed to be true, then P(a) is also true,where a is an arbitrary member of the domain of the discourse. This rule is called the universal specification (US)
If P(a) is true, where a is an arbitrary member of the domain of the discourse, then ∀x P(x) is true. This rule is called the universal generalization (UG)
If the statement ∃x P (x) is true, then P(a) is true, for some member of the domain of the discourse. This rule is called the existential specification (ES)
If P(a) is true for some member a of the domain of the discourse, then ∃x P(x) is also true. This rule is called the existential generalization (EG)
Discrete Mathematical Structures: Theory and Applications

65. Quantifiers and First Order Logic
Counterexample
An argument has the form ∀x (P(x ) → Q(x )), where the domain of discourse is D
To show that this implication is not true in the domain D, it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true
This means that there exists some x in D such that P(x) is true but Q(x) is not true. Such an x is called a counterexample of the above implication
To show that ∀x (P(x) → Q(x)) is false by finding an x in D such that P(x) → Q(x) is false is called the disproof of the given statement by counterexample
Discrete Mathematical Structures: Theory and Applications

66. 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
Discrete Mathematical Structures: Theory and Applications

67. 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
Discrete Mathematical Structures: Theory and Applications

68. 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
Discrete Mathematical Structures: Theory and Applications

69. 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
Discrete Mathematical Structures: Theory and Applications

70. Proof Techniques
Discrete Mathematical Structures: Theory and Applications

71. 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
Discrete Mathematical Structures: Theory and Applications

72. 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
Or, prove that p if and only if q, and then q if and only if r
Other methods are possible
Discrete Mathematical Structures: Theory and Applications

73. Algorithms
Definition: step-by-step problem-solving process in which a solution is arrived at in a finite amount of time
All algorithms have the following properties:
Input : For example, a set of numbers to find the sum of the numbers
Output : For example, the sum of the numbers
Precision : Each step of the algorithm is precisely defined
Uniqueness : Results of each step are unique and depend on the input and results of previous step
Finiteness : Algorithm must terminate after executing a finite number of steps
Generality : Algorithm is general in that it applies to a set of inputs
Discrete Mathematical Structures: Theory and Applications

74. Algorithms Pseudocode Conventions
The symbol := is called the assignment operator
Example: The statement x := a is read as “assign the value a to x” or “x gets the value a” or “copy the value of a into x”
x := a is also known as an assignment statement
Control Structures
One way-selection
if booleanExpression then statement
If booleanExpression evaluates to true, statement is evaluated
Two way-selection
if booleanExpression then statement1
else statement2
If booleanExpression evaluates to true , statement1 executes,
otherwise statement2 executes
Discrete Mathematical Structures: Theory and Applications

75. Algorithms Pseudocode Conventions Control Structures
The while loop takes the form:
while booleanExpression do loopBody
The booleanExpression is evaluated. If it evaluates to true, loopBody executes. Thereafter loopBody continues to execute as long as booleanExpression is true
The for loop takes the form:
for var := start to limit do loopBody
var is an integer variable. The variable var is set to the value specified by start. If var limit, loopBody executes. After executing the loopBody , var is incremented by 1. The statement continues to execute until var > limit
Discrete Mathematical Structures: Theory and Applications

76. Algorithms Pseudocode Conventions Control Structures
The do/while loop takes the form:
do loopBody while booleanExpression
The loopBody is executed first and then the booleanExpression is evaluated. The loopBody continues to execute as long as the booleanExpression is true
Discrete Mathematical Structures: Theory and Applications

77. Algorithms Pseudocode Conventions Block of Statement
To consider a set of statements a single statement, the statements are written between the words begin and end
begin
statement1
statement2
...
statementn;
end
Discrete Mathematical Structures: Theory and Applications

78. Algorithms Pseudocode Conventions Return Statement
The return statement is used to return the value computed by the algorithm and it takes the following form:
return expression;
The value specified by expression is returned. In an algorithm, the execution of a return statement also terminates the algorithm
Read and Print Statements
read x;
Read the next value and store it in the variable x
print x;
Output the value of x
Discrete Mathematical Structures: Theory and Applications

79. Algorithms Pseudocode Conventions Arrays (List)
A list is a set of elements of the same type
The length of the list is the number of elements in the list
L[1...n ]. L is an array of n components, indexed 1 to n . L[i ] denotes the ith element of L
For data in tabular form, a two-dimensional array is used:
M[1...m,1...n ] M is a two-dimensional array of m rows and n columns
The rows are indexed 1 to m and the columns are indexed 1 to n
M[i,j] denotes the (i,j)th element of M, that is, the element at the ith row and jth column position
Discrete Mathematical Structures: Theory and Applications

80. Algorithms Pseudocode Conventions Subprograms (Procedures)
In a programming language,an algorithm is implemented in the form of a subprogram, a.k.a. a subroutine or a module
Two types of subprograms
Functions
Returns a unique value
Procedure
Other types of subprograms
body of the function or procedure is enclosed between the words begin and end
the execution of a return statement in a function terminates the function
Discrete Mathematical Structures: Theory and Applications

81. Algorithms Pseudocode Conventions Comments
In describing the steps of an algorithm, comments are included wherever necessary to clarify the steps
Two types of comments: single-line and multi-line
Single-line comments start anywhere in the line with the pair of symbols //
Multi-line comments are enclosed between the pair of symbols /* and */
Specifies what the algorithm does, as well as the input and output
Discrete Mathematical Structures: Theory and Applications

82. Algorithms Polynomial Operations Polynomial in one variable is
ai are real (or complex) and n is a non-negative integer
If p(x) = a0 then p(x) is a constant polynomial
If p(x) is a nonzero constant polynomial, then the degree of p(x) is defined to be 0
If p(x) is not constant and an 0, then n is called the degree of p(x); that is, the degree of a non-constant polynomial is defined to be the exponent of the highest power of x
The basic operations performed on polynomials are to add, subtract, multiply, and divide polynomials, and to evaluate a polynomial at a given point
Discrete Mathematical Structures: Theory and Applications