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Tutorial 2: First O rder Logic and Methods of Proofs

This tutorial dives into the fundamentals of First Order Logic, covering key topics such as the order of quantifiers, formulations, and negation. You'll learn essential proof methods including direct proofs, contrapositive reasoning, and proof by contradiction. We also explore how to express mathematical statements using first order logic, analyze equivalence in quantifiers, and tackle classic problems like the pigeonhole principle. Whether you're new to logic or need a refresher, this session will equip you with the necessary tools for effective reasoning and proof construction.

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Tutorial 2: First O rder Logic and Methods of Proofs

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

  14. 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

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

  16. Contradiction • Show that is not 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!!!)

  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 • http://en.wikipedia.org/wiki/Pigeonhole_principle

  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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