Proof by Contradiction

Truth Table For “If…., then….”

Pattern

We want to proof “If A then B”

1. Assume A and ~B (A is True and B is False)
2. Derive (A is True and B is False) to a contradiction.
   1. Possibly contradicting to original assumptions
   2. or contradicting to facts, e.g. 2>1
3. We then proof “If A then B” by contradiction

Reason

1. The proposition “If A then B” is False only when A = True and B = False
2. Under condition either 1. or 2. , we have the conclusion that (A = True and B = False) is False and the complement of (A = True and B = False) always leads to “If A, then B” is True.
3. Thus, we then proof “If A then B” by contradiction. That is, (A \(\cap\) ~B = \(\phi\) ).