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_principle

Conclusion

• Contrapositive– Find the contrapositive form– Prove it

• Contradiction– Assume it is false– Show it is impossible by finding contradiction