 ## Presentation on theme: "PROOF BY CONTRADICTION"— Presentation transcript:

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.

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.

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.

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.

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.)