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

• You know that

Negation

• 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 n 2 is divisible by 3, then n is divisible by 3

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 3 is not divisible by 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 2 + 12k + 4 = 3(3k 2 + 4k + 1) + 1 Both are not divisible by 3

Contradiction • Show that is not rational.

– Given If n 2 is divisible by 3, then n is divisible by 3

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!!!)

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