Indirect Approaches

Sometimes proving something directly is tricky, so mathematicians use clever indirect methods.

Proof by Contradiction

Assume the opposite of what you want to prove, show this leads to a contradiction (something impossible), so your original statement must be true.

Proof by Contrapositive

Instead of proving "if A, then B," you prove "if not B, then not A." Since these are logically equivalent, this can be easier!

When to Use These Methods

• When a direct proof feels complicated or stuck.
• When the statement involves "impossible" or "never" outcomes.

Practice

These methods are powerful tools for tricky statements that seem hard to prove head-on.