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

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

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

3 First Order Logic

4 Order of quantifier Which one are equivalent?

5 Order of quantifier Which one are equivalent?

6 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

7 Formulation Express the following using first order logic

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

9 Negation Write down the negation of the following statements

10 Method of Proof

11 Direct Proof For every positive integer n, is even

12 Direct Proof For every positive integer n, is even

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

14 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

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

16 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!!!)

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

18 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

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

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