Download presentation

Presentation is loading. Please wait.

Published byDenzel Daw Modified over 3 years ago

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

Similar presentations

Presentation is loading. Please wait....

OK

PROOF BY CONTRADICTION

PROOF BY CONTRADICTION

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on teachers day speech Ppt on 9 11 attack on world trade center and effects Ppt on water cycle in hindi Ppt on job rotation benefits Ppt on game theory wiki Ppt on cartesian product multiplication Ppt on chapter 3 atoms and molecules worksheets Download ppt on phase controlled rectifiers Ppt on porter's five forces analysis Ppt on drinking water in rural areas