# Proof by Contradiction

** Published:**

## Truth Table For “If…., then….”

A | B | If A, then B |
---|---|---|

T | T | T |

T | F | F |

F | T | T |

F | F | T |

## Pattern

We want to proof “If A then B”

- Assume A and ~B (A is True and B is False)
- Derive (A is True and B is False) to a contradiction.
- Possibly contradicting to original assumptions
- or contradicting to facts, e.g. 2>1

- We then proof “If A then B” by contradiction

## Reason

- The proposition “If A then B” is False only when A = True and B = False
- 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.
- Thus, we then proof “If A then B” by contradiction. That is, (A \(\cap\) ~B = \(\phi\) ).