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

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google