Download presentation

1
**PROOF BY CONTRADICTION**

2
**proof by contradiction**

Let r be a proposition. A proof of r by contradiction consists of proving that not(r) implies a contradiction, thus concluding that not(r) is false, which implies that r is true.

3
**proof by contradiction**

In particular if r is if p then q then not(r) is logically equivalent to p AND (not(q)). This can be verified by constructing a truth table.

4
**proof by contradiction**

So a true proposition if p then q may be proved by contradiction as follows: Assume that p is true and q is false, and show that this assumption implies a contradiction.

5
**proof by contradiction**

One way of proving that the assumption that p is true and q is false implies a contradiction is proving that this assumption implies that p is false. Since the same assumption also implies that p is true, we conclude that the assumption implies that p is true and p is false, which is a contradiction.

6
**proof by contradiction**

EXAMPLE: Prove that the sum of an even integer and a non-even integer is non-even. (Note: a non-even integer is an integer that is not even.)

7
**proof by contradiction**

We have to prove that for every even integer a and every non-even integer b, a+b is non-even. This is the same as proving that For all integers a,b, if [a is even and b is non-even] then [a+b is non-even].

8
**proof by contradiction**

We prove that by contradiction. Assume that [a is even and b is non-even], and that [a+b is even]. So for some integers m,n, a=2m and a+b=2n. Since b=(a+b)-a, b=2n-2m=2(n-m). We conclude that b is even. This leads to a contradiction, since we assumed that b is non-even.

Similar presentations

Presentation is loading. Please wait....

OK

Copyright © 2007 Pearson Education, Inc. Slide 8-1.

Copyright © 2007 Pearson Education, Inc. Slide 8-1.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on porter's five forces analysis Ppt on porter's five forces example Ppt on social etiquettes Ppt on dot net Ppt on recycling of waste material in india Ppt on condition of girl child in india Ppt on area of parallelogram for class 9 Download ppt on algebra for class 6 Ppt on autonomous car Ppt on ideal gas law definition