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
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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.
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•
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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