Indirect Methods Summary

Intro to Proofs · Axiom Academy

SUMMARY Indirect Proof Methods How contrapositive and contradiction give you powerful alternatives when a direct approach seems difficult. Two powerful alternatives: when a direct proof is difficult, the contrapositive reverses the logical flow and contradiction explores impossible scenarios. Logical equivalence: the contrapositive is logically identical to the original statement — you prove the same thing from a different angle. Contradiction's power: assuming the opposite and finding an impossibility is especially effective for existence and irrationality proofs. Method selection: choose by problem structure — contrapositive for conditionals with clear negations, contradiction for complex existence claims. Practice recognition: spotting when an indirect method is advantageous is as important as executing it correctly. Core Concept Proof by Contrapositive To prove , instead prove its contrapositive . The two are logically identical, so proving one proves the other automatically — often turning a hard direct proof into a straightforward one. When to use: when negating q gives more concrete information than p provides. Watch out for: the contrapositive is , NOT the converse . Classic example: if n^2 is odd then n is odd — easier via contrapositive (if n is even, then n^2 is even). Core Concept Proof by Contradiction

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