Course Outline

Book: Discrete Mathematics by K. P. BogartTopics:

Sets and statementsSymbolic LogicRelations functionsMathematical InductionCounting TechniquesRecurrence relationsTreesGraphs

Grades: First: 25%Second 25% Final 50%*Note: The outline is subject to change

Discrete Mathematics

Is the one we use to analyze discrete processes that are carried out in a step-by-step fashion.

Algorithm

A list of step by step instructions for carrying out a process

Chapter 1

Sets and Statements

Statements

A declarative sentence can be true, false or ambiguous

A statement is an unambiguous declarative sentence that is either true or false

Example

5 plus 7 is 12 5 plus 7 is 5 5 plus 7 is large Did you have coffee this morning?

Sets

Set: an unambiguous description of a collection of objects

EX:

Set of outcomes for flipping a coin

S={H,T}

However, the list of outcomes might be:

HTTTHHH…….

Sets

Members of a set are called elements– aA “a is an element of A”

“a is a member of A”– aA “a is not an element of A”

EX: Set of +ve integersS={x |x>0}3 S-5 S

Sets

Universe of a statement is the set whose elements are discussed by the statement

EX:x multiplied by x is +veThe universe could be:- Set of +ve integers- Set of –ve integers- Set of all integersFlipping a coin-Universe: {H,T}

Sets

Note: P, q, r, s are used to represent statements X, y, z, w are used to represent variables

Compound Statements

Simple statements are represented by symbolsEX: P: x is a positive integer Compound statements are represented by symbols+ logical

connectivesLogical Connectives:

– Conjunction AND. Symbol ^ – Inclusive disjunction OR Symbol v– Exclusive disjunction OR Symbol (+)– Negation Symbol ¬– Implication Symbol

Compound Statements

Example:-I will take calculas1 and I will take physics class.Represented as: p ^ q- I will have coffee or I will have teaRepresented as: p v q- Ali is at school or Ali is at homeRepresented as: p (+) q- p: x is greater than 2 ¬p: x is not greater than 2-George is at school and either Sue is at store or Sue is at home.P ^( q (+) r )*Note the use of parentheses ( see example 4 page 7).

Truth sets

The set of all values of x that make a symbolic statement p(x) true is called the truth set of the proposition p.

(the set of all values in the universe that makes p true).

The symbolic statements p(x) & q(x) are equivalent if they have the same truth sets.

Truth sets

EX:Universe: The result of flipping 2 coins

P: the result has one head q: the result has one tail

P and q are equivalent since they have the same truth sets.

Fundamental Principle of Set Equality

To show that the sets T and S are equal, we may show that each element in T is an element in S and vice versa.

EX:Universe: 300 coin flipsP: the result has 2 H’sq: the result has 298 T’sShow that p and q are equivalent.

Finite and infinite sets

Finite sets - Examples:

A = {1, 2, 3, 4} B = {x | x is an integer, 1 < x < 4} D = {dog, cat, horse} D = {dog, cat, horse}

Infinite sets- Examples:

Z = {integers} = {…, -3, -2, -1, 0, 1, 2, 3,…} Natural numbers N = {0, 1, 2, 3, …} S={x| x is a real number and 1 < x < 4} = [0, 4]

Section 1.2: Sets

Venn diagrams

A Venn diagram provides a graphic view of sets and their operations: union, intersection, difference and complements can be identified

Set operations

Given two sets X and Y the following are operations that can be performed on them:– Union– Intersection– Complement– Difference

Union

The union of X and Y is defined as the set A B = { x | x A or x B}

Intersection

The intersection of X and Y is defined as the set: X Y = { x | x X and x Y}

Two sets X and Y are disjoint

if X Y =

XY

xy

XY

X Y =

Complement

The complement of a set Y contained in a universal set U is the set Yc = U – Y

YUYc

Difference

The difference of two sets

X – Y = { x | x X and x Y}

The difference is also called the relative complement of Y in X

X YX-y

Properties of set operations

Theorem : Let U be a universal set, and A, B and C subsets of U.

The following properties hold:a) Associativity: (A B) C = A (B C) (A B) C = A (B C)b) Commutativity: A B = B A A B = B A

Properties of set operations (2)

c) Distributive laws: A(BC) = (A B) (A C) A(BC) = (A B) (A C)

d) Identity laws: AU=A A = A

e) Complement laws: AAc = U AAc =

Properties of set operations (3)

f) Idempotent laws:

AA = A AA = A

g) Bound laws:

AU = U A =

h) Absorption laws:

A(AB) = A A(AB) = A

Properties of set operations (4)

i) Involution law: (Ac)c = A

j) 0/1 laws: c = U Uc =

k) De Morgan’s laws for sets:

(AB)c = AcBc

(AB)c = AcBc

Demorgan’s Laws for sets

~(A B) = (~A) (~B)

-Proof: To be discussed in class

~(A B) = (~A) (~B)

-Proof: exercise

Theorem

Let p and q be statements and let P and Q be their truth sets, then:

- P Q is the truth set of p^q (proof discussed in class)

- P Q is the truth set of pvq- ~P is the truth set of ¬p

Example: Venn Diagrams

Show that P (Q R) = (P Q) (P R)

Using Venn diagrams

- See example 9 page 18

Subsets

It is a relation between sets ( not operation) A set S is a subset of set T if each element in S is also an

element in T. Examples:

A = {3, 9}, B = {5, 9, 1, 3}, is A B ?

A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, is A B ?

A = {1, 2, 3}, B = {2, 3, 4}, is A B ?

Equality: X = Y if X Y and Y X

Subsets using Venn diagrams

The ellipse is a subset of the circle

Theorem

Let R and S be two sets then:

- R and S are subsets of R S- R S is a subset of both R and S- R S = S if and only if R S- R S=R if and only if R S

Example

Prove that

R (S T) S (R T)

The Empty Set

The empty set has no elements.

Also called null set or void set.

EX:

P is the truth set of p: x>0

Q is the truth set of q: x<0

The truth set of p^q = P Q= P and Q are disjoint sets

Section 1.3

Determining the Truth of Symbolic Statements

Truth tables

Truth tables are used to determine truth or falsity of compound statements

Truth table of conjunction

Truth table of conjunction

p ^ q is true only when both p and q are true.

p q p ^ q

T T T

T F F

F T F

F F F

Truth table of disjunction

p q is false only when both p and q are false

p q p v q

T T T

T F T

F T T

F F F

Exclusive disjunction

p (+) q is true only when p is true and q is false, or p is false and q is true. Example: p = "John is programmer, q = “John is a lawyer" p (+) q = "Either John is a programmer or John is a lawyer"

p q p (+) q

T T F

T F T

F T T

F F F

Negation

Negation of p: in symbols ¬p

¬ p is false when p is true, ¬ p is true when p is false Example: p = "John is a programmer" ¬ p = "It is not true that John is a programmer"

p ¬ p

T F

F T

Truth tables

Examples:

Truth table for :- ¬pvq- (pvq) ^ ¬(p^q)

Definition

2 statements are equivalent if their truth tables have the same final column

Exercise

Use the truth tables to find out whether the following statements are equivalent:

- (p^q) v (p^r)- P^(qvr)

Section 1.4

The Conditional Connectives

Conditional propositions and logical equivalence

A conditional proposition is of the form “If p then q” In symbols: p q Example:

– p = " John is a programmer"– q = " Mary is a lawyer "– p q = “If John is a programmer then Mary is a

lawyer"

Truth table of p q

p q is true when both p and q are true

or when p is false

p q p q

T T T

T F F

F T T

F F T

P q is equivalent to ¬pvq

Recall: 2 statements are equivalent if their truth tables have the same final column

Exercise:Show that p q and ¬p v q are equivalent.

Note: it is important to represent the implication() and the exclusive OR(+) using other connectives (^,V, ¬), why??

Example

Rewrite without arrows:

¬r ( s v (r ^ t))

Example

Consider flipping a coin 3 times p is the statement “ the first flip comes up

heads” q is the statement “there are at least 2

heads”

Find the truth sets of p, q, pq

Answer: {TTT,TTH,THT,THH,HHH,HHT,HTH}

Section 1.5

Boolean Algebra:

When we apply known laws about set operations to derive other ones algebraically, we say we are doing Boolean Algebra.

Example: ( not required)

Use Boolean algebra to prove the unique inverse property. if x P= and x P = U then x= ~Px = x U (identity law) = x (P ~P) (inverse law) = (x P) (x ~P) (distributive law) = (x ~P) (given property) = (P ~P) (x ~P) (Inverse law) = (P x) ~P (distributive law) = U ~P (given property) = ~P (Identity law)

Boolean Algebra for statements

A formula says that 2 truth sets are equal corresponds to a formula saying that 2 statements are equivalent ( so all set laws are translated directly into statement laws).

The statements about a universe satisfy the following rules: a) Associativity: (p V q) V r = p v (q v r) (p ^ q) ^ r = p ^ (q^ r)

b) Commutativity: p V q = q V p p ^ q = q ^ p

Boolean Algebra for Statements

c) Distributive laws: p ^ (q v r) = (p ^ q) V (p ^ r) p V ( q ^ r) = (p V q) ^(p V r)

d) Identity laws: p^1=p pV0 = p

e) Complement laws: p V ¬p = 1 p ^ ¬p = 0f) Idempotent laws: p V p = p p ^ p = p

g) Bound laws: p V 1 = 1 p ^ 0 = 0

h) Absorption laws:p v ( p ^ q ) = p p ^ ( p v q) = p

i) Double negation law: ¬ ¬p = p

j) De Morgan’s laws:

¬(p V q) = ¬ p ^ ¬ q

¬(p ^ q) = ¬ p V ¬q

Final Example

Simplify:- (¬ ¬r) V (s V (r ^ t))

Answer : r V s

- (¬ (r ^ s) V (r V s)) ^ (¬ (r V s) V (r ^ s))

Answer: (¬r ^ ¬s ) V (r ^ s)