Assume the opposite

The idea

To prove a statement by contradiction, assume to the contrary that it is false and reason until a contradiction arises; the contradiction shows the assumption was incorrect, so the statement is true.

No largest even number

Meera says: “There is no largest even number.” We cannot check all even numbers, there are too many.

How can we be sure she is right?

• Check bigger and bigger even numbers one by one
• Pretend there IS a largest one and see what goes wrong
• Just trust her

Climb the ladder

Climb the proof rung by rung. At the end, decide what must be wrong.

Proof by contradiction

To prove a statement by contradiction, assume to the contrary that it is false, and reason until a contradiction arises. The contradiction shows the assumption was incorrect, so the statement is true.

The four moves: 1. Assume the opposite. 2. Reason correctly, step by step. 3. Reach a statement that clashes with something true. 4. Conclude: the assumption was wrong, so the original statement is true.

The steps were correct and the known facts were true. The only thing left to blame is the assumption.

Notes

To prove by contradiction: assume the statement is false, reason until a contradiction arises, then conclude the assumption was incorrect, so the statement is true.

Check yourself

In a proof by contradiction, what do we do first?

A proof ends: “This contradiction has arisen because of our incorrect assumption.” What do we conclude?

To prove “no whole number is both even and odd” by contradiction, what do we assume?

• Assume the statement we want to prove is false — correct. Yes! We assume the opposite and look for a clash.
• Assume the statement is true. Then there is nothing to prove. We assume the opposite.
• Check a few examples. Examples can never prove a statement for all cases.
• The original statement is true — correct. Yes! The assumed opposite is false, so the statement is true.
• The original statement is false. The assumption was the OPPOSITE of the statement. It is the opposite that was wrong.
• We still do not know. The clash tells us the assumption is wrong, and that settles it.
• Every whole number is both even and odd. The opposite of “no number is …” is “some number is …”.
• Some whole number is both even and odd — correct. Yes! The opposite of “none” is “at least one”.
• No whole number is even. That is a different statement.