Discrete Structures – CS2300

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Text

Discrete Mathematics and Its Applications

Kenneth H. Rosen (7th Edition)

Chapter 1

The Foundations: Logic and Proofs

About This Course

• The Conceptual Foundation of Computer Science

• Prerequisite for CS 3240 (Theory of Computation)

• Applied Mathematics Course

Continuous vs. Discrete Math

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Continuous Discrete

Sliding down a slidePouring water

Length of ropeCrawling slug

Adding milkGrade point average

Climbing up stairsStacking ice cubesNumber of knotsHopping rabbitAdding eggsCalculus grade

Discrete Solutions

• How many ways are there to choose a valid password?

• What is the probability of winning the lottery?• Is there a path linking two particular computers

in a network?• What is the shortest path between two

destinations using a transportation system?• How many valid Internet addresses are there?

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Chapter 1 Objective

“In this chapter we will explain what makes up a correct mathematical [logical] argument and introduce tools to construct these arguments.”

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Sections 1.1, 1.2

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Logic

Propositional Logic

Propositions

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

Six is a prime number.

Count is less than ten.

7<5

Consider this statement.

Compound Propositions

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Compound propositions are formed from existing propositions using logical operatorsToday is Wednesday and it is snowing outside.

12 is not a prime number.

Negation of a Proposition

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T

F

F

T

NOT !P P

Negation of a Proposition

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repeat{…}until(feof(my_file));

while (!feof(my_file)){…}

Disjunction of Two Propositions

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T T

T F

F T

F F

T

T

T

F

OR ||qp p q

Disjunction of Two Propositions

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repeat{ …}until(count>10 || feof(myfile));

if(choice==PAUSE || choice ==STOP) ...

Conjunction of Two Propositions

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T T

T F

F T

F F

T

F

F

F

AND &&p q qp

Conjunction of Two Propositions

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while(!feof(a_file) && index<SIZE){ …}

if(!done && time_left) ...

Exclusive-OR of Two Propositions

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T T

T F

F T

F F

F

T

T

F

Exactlyone ofthem istrue.

p q qp ^

“but not both”

Implication

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T T

T F

F T

F F

T

F

T

T

p is called thehypothesis and q is theconclusion

p q qp

Implication (“Conditional”)

• “if p, then q”

• “p implies q”

• “if p,q”

• “p only if q”

• “p is sufficient for q”

• “q if p”

• “q whenever p”

• “q is necessary for p”

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T T

T F

F T

F F

T

F

T

T

p q qp

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q whenever p

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T T

T F

F T

F F

T

F

T

T

Suppose that the proposition is true. Then, q is true whenever p is true.

p q qp

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p is sufficient for q

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T T

T F

F T

F F

T

F

T

T

Suppose that the proposition is true. Then, to guarantee that q is true it is sufficient to say that p is true.

p q qp

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Converse of an Implication

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T T

T F

F T

F F

T

F

T

T

T

T

F

T

p q qp AndConversely

qp

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Example of Converse

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If it stays warm for a week, the apple trees will bloom.If the apple trees bloom, it will be warm for a week.

If x is even then x2 is even.

If x2 is even then x is even.

Contrapositive of an Implication

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T T

T F

F T

F F

T

F

T

T

T

F

T

F F

F T

T F

T T T

p qqp pq pq

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Examples of Contrapositive

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If it snows tonight, then I will stay at home.

If I do not stay at home, then it didn’t snow tonight.

If x is odd then x2 is odd.

If x2 is not odd then x is not odd.

If x2 is even then x is even.

Biconditional

T T

T F

F T

F F

T

F

T

T

T

T

F

T

T

F

F

T

p qpq qp )()( qpqp

qp

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Biconditional

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pif and only if q p iff q

qp )()( qpqp

Bitwise operators

1101 10011110 01001100 0000

AND1101 10011110 01001111 1101

OR

1101 10011110 01000011 1101

XOR

a&b a|b

a^b

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t01_1_009.jpg

Tautology

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Tautology - a compound proposition that is always true.

T T T TT F F TF T T TF F T T

p qpqpqp )(

pqp )(

Contradiction

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Contradiction - a compound proposition that is always false.

T F F

F T F

p pp p

Contingency

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A contingency is neither a tautology nor a contradiction.

T T T TT F F FF T F TF F F T

p qp q )( qpp )( qpp