Discrete Structures – CS Text Discrete Mathematics and Its Applications Kenneth H. Rosen (7 th 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 3 ContinuousDiscrete Sliding down a slide Pouring water Length of rope Crawling slug Adding milk Grade point average Climbing up stairs Stacking ice cubes Number of knots Hopping rabbit Adding eggs Calculus 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? 4

Chapter 1 Objective “In this chapter we will explain what makes up a correct mathematical [logical] argument and introduce tools to construct these arguments.” 5

Sections 1.1, Logic Propositional Logic

Propositions 7 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 8 Compound propositions are formed from existing propositions using logical operators Today is Wednesday and it is snowing outside. 12 is not a prime number.

Negation of a Proposition 9 TFTF F T NOT !

Negation of a Proposition 10 repeat{ … }until(feof(my_file)); while (!feof(my_file)){ … }

Disjunction of Two Propositions 11 T T F F T F T T T F OR ||

Disjunction of Two Propositions 12 repeat{ … }until(count>10 || feof(myfile)); if(choice==PAUSE || choice ==STOP)...

Conjunction of Two Propositions 13 T T F F T F T F F F AND &&

Conjunction of Two Propositions 14 while(!feof(a_file) && index<SIZE) { … } if(!done && time_left)...

Exclusive-OR of Two Propositions 15 T T F F T F F T T F Exactly one of them is true. ^ “but not both”

Implication 16 T T F F T F T F T T p is called the hypothesis and q is the conclusion

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” 17 T T F F T F T F T T 17

q whenever p 18 T T F F T F T F T T Suppose that the proposition is true. Then, q is true whenever p is true. 18

p is sufficient for q 19 T T F F T 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. 19

Converse of an Implication 20 T T F F T F T F T T T T F T And Conversely 20

Example of Converse 21 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 x 2 is even. If x 2 is even then x is even.

Contrapositive of an Implication 22 T T F F T F T F T T T F T F F F T T F T T T 22

Examples of Contrapositive 23 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 x 2 is odd. If x 2 is not odd then x is not odd. If x 2 is even then x is even.

Biconditional T T F F T F T F T T T T F T T F F T 24

Biconditional 25 p if and only if qp iff q

Bitwise operators AND OR XOR a&b a|b a^b 27

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Tautology 29 Tautology - a compound proposition that is always true. T T T T T F F T F T T T F F T T

Contradiction 30 Contradiction - a compound proposition that is always false. T F F F T F

Contingency 31 A contingency is neither a tautology nor a contradiction. T T T T T F F F F T F T F F F T