1. Tutorial 2: First Order Logic and Methods of Proofs
Peter Poon

2. Agenda First Order Logic Methods of Proofs Order of quantifier
Formulation
Negation
Methods of Proofs
Direct Proof
Contrapositive
Contradiction

3. First Order Logic

4. Order of quantifier
Which one are equivalent?

5. Order of quantifier
Which one are equivalent?

6. 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”

7. Formulation
Express the following using first order logic

8. Negation
You know that
Write down the negation of the following statements

9. Negation
Write down the negation of the following statements

10. Method of Proof

11. Direct Proof
For every positive integer n, is even

12. Direct Proof
For every positive integer n, is even

13. Contrapositive
If n2 is divisible by 3, then n is divisible by 3

14. 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

15. Contradiction Show that is not rational.
Given If n2 is divisible by 3, then n is divisible by 3

16. Contradiction Show that is not rational. If is 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!!!)

17. Contradiction
If there 40 pigeons sharing 7 pigeonholes, then at least 1 pigeonhole have more then 5 pigeons.

18. 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

19. Conclusion Contrapositive Contradiction Find the contrapositive form
Prove it
Contradiction
Assume it is false
Show it is impossible by finding contradiction