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Core explanation

Not every statement is easy to prove directly. Sometimes the conclusion is difficult to reach from the assumption — but the negation of the conclusion leads quickly to something impossible.

This is where indirect proofs come from.

There are two major forms:

Contrapositive Proof
- Instead of proving "If P, then Q," prove the logically equivalent statement:
  
  If not Q, then not P.
- Often much cleaner than direct proofs.
Proof by Contradiction
- Assume the statement you want to prove is false.
- Show that this assumption leads to a contradiction.
- Conclude the statement must be true.

Indirect proofs exist because they can turn a messy direct argument into a clean, natural one by reframing the logic.

Example:

To prove "If n² is even, then n is even," the contrapositive is far easier: If n is odd, then n² is odd.

TL;DR — key idea

Indirect proofs reframe a hard proof into an easier one by proving an equivalent or stronger contradiction-based statement.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
Explain why indirect proofs are needed. Give an example of a statement where the direct proof is awkward, but the contrapositive is simple.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Why Indirect Proof Exists

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