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Discrete Structures – CS2300. 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)
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Discrete Structures – CS2300 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 Continuous Discrete Sliding down a slide Climbing up stairs Pouring water Stacking ice cubes Length of rope Number of knots Crawling slug Hopping rabbit Adding milk Adding eggs Grade point average 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?
Chapter 1 Objective “In this chapter we will explain what makes up a correct mathematical [logical] argument and introduce tools to construct these arguments.”
Sections 1.1, 1.2 Logic Propositional Logic
Propositions 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 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 T F NOT F T
Negation of a Proposition repeat{…}until(feof(my_file)); while (!feof(my_file)){…}
T T T F F T F F || Disjunction of Two Propositions OR T T T F
Disjunction of Two Propositions repeat{ …}until(count>10 || feof(myfile)); if(choice==PAUSE || choice ==STOP) ...
T T T F F T F F && Conjunction of Two Propositions AND T F F F
Conjunction of Two Propositions while(!feof(a_file) && index<SIZE){ …} if(!done && time_left) ...
T T T F F T F F ^ Exclusive-OR of Two Propositions F Exactlyone ofthem istrue. T T F “but not both”
T T T F F T F F Implication p is called thehypothesis and q is theconclusion T F T T
T T T F F T F F T F T T 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 T F F T F F T F T T q whenever p Suppose that the proposition is true. Then, q is true whenever p is true. 18
T T T F F T F F T F T T p is sufficient for q Suppose that the proposition is true. Then, to guarantee that q is true it is sufficient to say that p is true. 19
AndConversely T T T F F T F F T F T T Converse of an Implication T T F T 20
Example of Converse 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.
T T T F F T F F F F F T T F T T T F T T Contrapositive of an Implication T F T T 22
Examples of Contrapositive 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.
T T F T Biconditional T T T F F T F F T T F F T F T T
Biconditional pif and only if q p iffq
AND OR 1101 10011110 0100 1101 10011110 0100 1100 0000 1111 1101 XOR 1101 10011110 0100 0011 1101 Bitwise operators a|b a&b a^b
Tautology Tautology - a compound proposition that is always true. T T T TT F F TF T T TF F T T
Contradiction Contradiction - a compound proposition that is always false. T F F F T F
T T T TT F F FF T F TF F F T Contingency A contingency is neither a tautology nor a contradiction.