Discrete Mathematics

• Goals of a Discrete Mathematics Learn how to think mathematically

1. Mathematical Reasoning

Foundation for discussions of methods of proof

2. Combinatorial Analysis

The method for counting or enumerating objects

3. Discrete Structure

Abstract mathematical structures used to represent discrete objects and relationship between them

• What will we learn from Discrete Mathematics

4. Algorithms Thinking

Algorithm is the specification for solving problems.

It’s design and analysis is a mathematical activity.

5. Application and Modeling

Discrete Math has applications to most area of study.

Modeling with it is an extremely important

problem-solving skill .

• How to learn Discrete Mathematics?

Do as many exercises as you possibly can !

Chapter 1 The Foundations: Logic, Sets, and Functions• Rules of logic specify the precise meaning of

mathematics statements.

• Sets are collections of objects.

• A function sets up a special relation between two sets.

1.1 Logic

Propositions

A proposition is a statement that is either true or false, but not both.

Propositions

1. This class has 25 students.

2. 4+8=12

3. 5+3=7

Not propositions

1. What time is it?

2. Read this carefully.

3. x+1= 2.

Examples

Definition 1. Let p be a proposition. The statement

“It is not the case of p” is a proposition, called the

negation of p and denoted by .p

• We let propositions be represented as p,q,r,s,….

The value of a proposition is either T(true) or F(false).

p: Toronto is the capital of Canada.

Canada. of capital not the is Toronto:p

Examples

Table 1. The Truth Table for the negation of a proposition

p

TF

FT

p

called connectives

Definition 2. Let p and q be proposition ｓ .The proposition “p and q”, denoted by , is the proposition that is true when both p and q are true and is false otherwise. The proposition is called the conjunction of p and q.

qp

qp

Table 2. The Truth Table for the conjunction of two propositions

p q

T TT FF TF F

TFFF

qp

Examples

today.raining isit andFriday isToday :

today.raining isIt :

Friday. isToday :

qp

q

p

　

Definition 3. Let p and q be proposition ｓ .The proposition “p or q”, denoted by , is the proposition that is false when both p and q are false and is true otherwise. The proposition is called the disjunction of p and q.

qp

qp

Table 3. The Truth Table for the disjunction of two propositions

p q

T TT FF TF F

TTTF

qp

Examples

today.raining isit or Friday isToday :

today.raining isIt :

Friday. isToday :

qp

q

p

　

Definition 4. Let p and q be proposition ｓ .The exclusive of p and q, denoted by , is the proposition that is true when exactly one of p and q is true and it is false otherwise.

qp

Table 4. The Truth Table for the exclusive or of two propositions

p q

T TT FF TF F

FTTF

qp

Examples

both.not but children or parents areThey :

children. areThey :

parents. areThey :

qp

q

p

　

Definition 5. Let p and q be proposition ｓ .The implication is the proposition that is false when p is true and q is false and true otherwise ， where p is called hypothesis and q is called the conclusion.

qp

Table 5. The Truth Table for the implication

p q

T TT FF TF F

TFTT

qp

Examples

beach. the togo will then weay,sunday tod isit if :

beach. the togo will we:

.sunday isIt :

qp

q

p

　

“If p, then q” or “ p implies q”.

Another example: If today is Friday, then 2+3=6.

Definition 6. Let p and q be proposition ｓ .The biconditional is the proposition that is true when p and q have the same truth values and is false otherwise.

qp

Table 6. The Truth Table for the biconditional

p q

T TT FF TF F

TFFT

qp

Examples

ay.sunday tod isit ifonly and ifbeach the togo will we:

ay.sunday tod isit :

beach. the togo willwe:

qp

q

p

“p if and only if q”.

Translating English Sentences into Logical Expressions　Example 1

You can access the Internet from campus only if you are a computer science major or you are not a freshman.

a . You can access the Internet from campus.

c. You are a computer science major.

f. You are freshman.

The sentence can be represented as )( fca

Example 2

You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.

q. You can ride the roller coaster.

r. You are under 4 feet tall.

s. You are older than 16 years old.

The sentence can be represented as qsr )(

Logic and Bit Operations

• A bit has two values: 1(true) and 0(false).

• A variable is called a Boolean variable if　 its value is either true or false.

• Bit operations are written to be AND, OR and XOR in programming languages.

,,

Table 7. Table for the bit operations OR,AND and XOR

x0011

y0101

0111

0001

0110

yx yx yx

Example

Extend bit operations to bit strings.

01 1011 0110

11 0001 1101

11 1011 1111 bitwise OR

01 0001 0100 bitwise AND

10 1010 1011 bitwise XOR

1.2 Propositional Equivalences

Definition 1. A tautology is a compound proposition that is always true no matter what the values of the propositions that occur in it. A contradiction is a compound proposition that is always false． A contingency is a proposition that is neither a tautology nor a contradiction.

Table 1. Examples of a Tautologyand a Contradiction.

pTF

FT

TT

FF

p pp pp

a contradiction

a tautologyExample 1.

Logic Equivalences

Definition 2. The proposition p and q are called logically equivalent if is a tautology. The notation denotes that p and q are logically equivalent.

qp qp

• Using a truth table to determine whether two propositions are equivalent

T T T T F F F

F F T F T T F

F T F F T F T

F F F F T T T

and for esTruth tabl 2. Table

qpq p q) (pq p q p

qp

q)(p

T T T F F

T T T T F

F F F F T

T T F T T

and

for esTruth tabl 3. Table

qpqpp p q

qp

qp

　

Example 2 Example 3equivalent equivalent

• Some important

equivalences.

qpq)(p

laws sorgan' DeM qpq)(p

r)(q)p(r)(qp

laws tive Distribur)(q)p(r)(qp

r)q(prq)(p

laws eAssociativ )(prq)(p

pqqp

laws eCommutativ pqqp

law negtion Double pp)(

ppp

LawsIdempotent ppp

FFp

Lawsion Dominat TTp

pFp

lawIdentity pTp

Name eEquivalenc

nces. Equivalel Logica5 Table

p

p

rq

q)p(q)(p

Fpp

Tpp

anlences. Equiv

Logical UsefulSome 6 Table

Example 4

.equivalent

logically are and

that Show

qp

q)) p((p

qp

q)p(F

q)p(p)p(

q)(pp

q)p(pq))p((p

Solution

:

Example 5

. tautologya is

that Show q) (pq)(p

T

TT

)()(

)()(

)()()()(

:

qqpp

qpqp

qpqpqpqp

Solution

1.3 Predicates and Quantifiers

Propositional function

A statement involving a variable x is P(x) is said to be a propositional function if x is a variable and P(x) becomes a proposition when a value has been assigned to x.

In general, a statement involving the n variables ).,...,,p(x as denoted is ,...,, 2121 nn xxxxx

Example 1

Let P(x) denote the statement “x>3”. What are the truth values of P(4) and P(2)?

Example 2

Let Q(x,y) denote the statement “x=y+3”. What are the truth values of the propositions Q(1,2) and Q(3,0)?

Quantifiers

The universal quantification of P(x),denoted as is the proposition “P(x) is true for all values of x

in the universe of discourse.”

)(xxP

Solution P(x): x has studied calculus.

S(x): x is in this class.

The statement can be expressed as ))()(( xpxSx

universal quantifier

true.is :Solustion xP(x)

Example 3

Express the statement “Every student in this class has studied calculus.

numbers? real ofset the

is discourse of universe the where,tion quantifica the

of ue truth val theis What ".1"number thebe Let

4 Example

xP(x)

xxp(x)

existence quantifier

The existential quantification of P(x),denoted as is the proposition “There exists an element x in the universe of discourse such that P(x) is true.”

)(xxP

".4" when instance,for - trueis 3"" since trueis )( :Solution x xxxP

false. is 4 since false is

.4321n conjunctio theas same theis :Solution

)P(xP(x)

)P()P()P()P(xP(x)

?"4 exceedingnot integers positive theof consists discourse of universe theand

"10"statement theis where, of ue truth val theis What 5 Example 2 xP(x)xP(x)

numbers? real ofset the

is discourse of universe thewhere,tion quantifica the

of ue truth val theis What ".3"statement thedenote Let

6 Example

xP(x)

xP(x)

true.is 4 since trueis

.4321n conjunctio theas same theis :Solution

)P(xP(x)

)P()P()P()P(xP(x)

Solution: Every student in your school has a computer or has a friend

who has a computer.

Solution: There is a student none of whose friends are also friends with

each other.

?"4 exceedingnot integers

positive theof consists discourse of universe theand "10"

statement theis where, of ue truth val theis What 7 Example2

x

P(x)xP(x)

school.your in

students all ofset theisy andboth x for discouse of universe the

and "friends, are and " is ,computer" a has " is where

English, into statement theTranslate

8 Example

yxF(x,y) xC(x)

F(x,y)))y(C(y)x(C(x)

school.your in students all ofset theis and for

discourse of universe theand friends are and means whereEnglish, into

statement theTranslate

9 Example

zx, y

baF(a,b)

F(y,z)))z))(yF(x,z)z(((F(x,y)yx

Translating Sentences into Logical Expressions

Example 10

Express the statements “Some student in this class has visited Mexico” and “every student in this class has visited either Canada or Mexico using quantifiers.

issolution The Canada." visitedhas "statement the and

Mexico" visitedhas"statement thebe Let :

M(x)).(C(x)

xC(x)

x M(x)Solution

Example 11

Express the statement “Everyone has exactly one best friend” as a logical expression.

. issolution The

." of friendbest theis "statement thebe Let :

B(x,z))y)((zz(B(x,y)yx

xyB(x,y)Solution

Example 12

Express the statement “There is a woman who has taken a flight on every airline in the world.

ly.respectiveairlines, all and airplanes, all world,in the

woman theall of consisits and ,for discourse of universes the

where, is solutition The

."on flght is " be and " takenhas " be Let

:

aw, f

Q(f,a)f(P(w,f)aw

afQ(f,a) fwP(w,f)

Solution

Negations: the negation of quantified expressions.

.)2(

. (1)

P(x)xxP(x)

P(x)xxP(x)

Example 14

There is a student in the class who has taken a course in calculus.

Example 13

Every student in the class has taken a course in calculus.