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Learn/indirect proofs/When NOT to Use Contradiction

Lesson subsection

When NOT to Use Contradiction

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

Although contradiction is powerful, it is not always the right tool.

Avoid contradiction when:

  1. A direct or contrapositive proof is cleaner. If a simple algebraic argument works, do not complicate it.

  2. The contradiction is manufactured rather than natural. If you must force complicated steps to produce a contradiction, you're using the wrong method.

  3. The negation of the statement is messy. Some statements have negations that introduce unnecessary complexity.

  4. You can construct a direct example or inequality.

Heuristics:

  • Use contrapositive for implications where negating the hypothesis and conclusion is clean.
  • Use contradiction for existence, minimality, parity, or rationality arguments.
  • Use direct proof when definitions directly apply.

Great problem solvers choose the method that makes the proof shortest and clearest.

TL;DR — key idea

Use contradiction only when it simplifies the argument — not when it complicates it unnecessarily.

Try these in your notebook

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

  • 1

    Pick any statement from Modules 1–4 and decide which proof technique fits best: direct, contrapositive, or contradiction. Justify *why* your choice is the most natural.

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

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.