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**Tutorial 2: First Order Logic and Methods of Proofs**

Peter Poon

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**Agenda First Order Logic Methods of Proofs 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 n2 is divisible by 3, then n is divisible by 3

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**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

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**Contradiction Show that is not rational.**

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

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**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!!!)

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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

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**Conclusion Contrapositive Contradiction Find the contrapositive form**

Prove it Contradiction Assume it is false Show it is impossible by finding contradiction

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1 Section 1.1 A Proof Primer A proof is a demonstration that some statement is true. We normally demonstrate proofs by writing English sentences mixed.

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