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Tutorial 2: First Order Logic and Methods of Proofs Peter Poon Agenda • First Order Logic – 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. – 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_principl e Conclusion • Contrapositive – Find the contrapositive form – Prove it • Contradiction – Assume it is false – Show it is impossible by finding contradiction