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

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

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First Order Logic

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Order of quantifier Which one are equivalent?

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Order of quantifier Which one are equivalent?

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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

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Formulation Express the following using first order logic

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

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Negation Write down the negation of the following statements

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Method of Proof

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Direct Proof For every positive integer n, is even

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Direct Proof For every positive integer n, is even

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Contrapositive If n 2 is divisible by 3, then n is divisible by 3

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

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Contradiction Show that is not rational. – Given If n 2 is divisible by 3, then n is divisible by 3

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Contradiction Show that is not rational. – Given If n 2 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 p 2 = 3q 2 so 9k 2 = 3q 2 q 2 = 3k 2 (p and q have the common factor 3 contradiction!!!)

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

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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 e e

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

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