Tutorial 2: First O rder Logic and Methods of Proofs
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Transcript of Tutorial 2: First O rder Logic and Methods of Proofs
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