Indirect Proof (Proof by Contradiction)

Indirect Proof (Proof by Contradiction)

How to solve indirect proof (proof by contradiction) problems: example and its solution.

How to write?

Start from assuming that '~prove' is true. Then show a contradiction.

Indirect proof is a way to prove a statement
when writing two column proof.

Start from assuming that '~prove' is true.

Then show a contradiction.

If there's a contradiction,
the first assumption ('~prove') must be the cause.

So '~prove' is false,
which means 'prove' is true.

Example

Given: AM is not equal to MB. Prove: M is not the midpoint of the line segment AB.

Assume that '~prove' is true:
M is the midpoint of AB.

Try to find a contradiction.
Start from the assumption.

AMMB (by the property of a midpoint)

AM = MB (definition of congruent segments)

AM = MB is derived.
But AMMB. (given)

Here's a contradiction.

This contradiction is caused by the assumption: '~prove' is true

So '~prove' is false.
And its negation, 'prove' is true.
So 'M is not the midpoint of AB' is true.