Common Pitfalls

Intro to Proofs · Axiom Academy

EXAMPLE Common Pitfalls in Indirect Proofs Identify and avoid critical errors in proof by contradiction and contrapositive arguments The Theorem (and a Flawed Proof of It) A student turns in the proof below. The theorem is true , but the student's argument is not valid . Work through it and pinpoint each error. Key Takeaways: Avoiding Common Pitfalls You found three critical errors in this indirect proof: Pitfall #1 — Assuming what you're trying to prove. In a proof by contradiction of "if then ," you assume together with the negation of . Here the student assumed is even (the conclusion) instead of is odd. Pitfall #2 — A "contradiction" that isn't one. Because the bad assumption was baked in, the algebra only re-derived the hypothesis ( is even). A valid contradiction proof must reach a genuine logical impossibility, not restate a premise. Pitfall #3 — Confusing contrapositive with converse. For "if then ": Contrapositive: "if not , then not " — logically equivalent. Converse: "if , then " — not equivalent. Prove: if is even, then is even. Contrapositive: if is odd, then is odd. Proof: assume is odd, so for some integer . Then which is odd. Since the contrapositive holds, the original statement holds. Set up your assumptions correctly, make sure your contradiction is real, and never mistake the converse for the contrapositive — these habits are what make a proof rigorous.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.