Tutorial 2: First Order Logic and Methods of Proofs

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Presentation transcript:

Tutorial 2: First Order Logic and Methods of Proofs Peter Poon

Agenda First Order Logic Methods of Proofs Order of quantifier Formulation Negation Methods of Proofs Direct Proof Contrapositive Contradiction

First Order Logic

Order of quantifier Which one are equivalent?

Order of quantifier Which one are equivalent?

Formulation Express the following using first order logic Let be the set of all positive integers be the set of all real numbers be “x is prime”

Formulation Express the following using first order logic

Negation You know that Write down the negation of the following statements

Negation Write down the negation of the following statements

Method of Proof

Direct Proof For every positive integer n, is even

Direct Proof For every positive integer n, is even

Contrapositive If n2 is divisible by 3, then n is divisible by 3

Contrapositive If n2 is divisible by 3, then n is divisible by 3 Contrapositive form If n is not divisible by 3, then n2 is not divisible by 3 Case 1: n = 3k + 1 n2 = (3k + 1)2 = 9k2 + 6k + 1 = 3(3k2 + 2k) + 1 Case 2: n = 3k + 2 n2 = (3k + 2)2 = 9k2 + 12k + 4 = 3(3k2 + 4k + 1) + 1 Both are not divisible by 3

Contradiction Show that is not rational. Given If n2 is divisible by 3, then n is divisible by 3

Contradiction Show that is not rational. If is rational Given If n2 is divisible by 3, then n is divisible by 3 If is rational Since , which is divisible by 3 So p = 3k, k is positive integer Also p2 = 3q2 so 9k2 = 3q2 q2 = 3k2 (p and q have the common factor 3 contradiction!!!)

Contradiction If there 40 pigeons sharing 7 pigeonholes, then at least 1 pigeonhole have more then 5 pigeons.

Contradiction If there 40 pigeons sharing 7 pigeonholes, then at least 1 pigeonhole have more then 5 pigeons. Assume it is false Then every pigeonhole have at most 5 pigeons Total number of pigeons <= 5 * 7 = 35 Contradiction!!! Pigeonhole principle http://en.wikipedia.org/wiki/Pigeonhole_principle

Conclusion Contrapositive Contradiction Find the contrapositive form Prove it Contradiction Assume it is false Show it is impossible by finding contradiction