The Method of Cases

Intro to Proofs · Axiom Academy

A fundamental proof technique: break a problem into distinct cases, prove each separately, then combine for one complete proof. When proving a statement, if you can split all possibilities into distinct cases and prove the statement holds in each case, then you have proven it for everything — no possibility is left out. Cover every possibility with cases, prove each, conclude P Show every scenario falls into at least one case. Prove P in case C₁, then C₂, … (one case at a time). Conclude: P holds in all cases, therefore P is true. Even / odd — split integers by parity. Positive / negative / zero — split by sign. Piecewise definitions — e.g. |x| behaves differently for vs. x < 0 . 2. Requirement #1: Exhaustive Cases Your cases must be exhaustive : together they must cover every possible scenario. If any situation slips through the cracks, your proof has a gap — even if each case you wrote is perfect. A non-exhaustive split (invalid) Trying to prove something about all integers using only: • Case 1: n > 0 • Case 2: n < 0 Problem: what about n = 0 ? It belongs to neither case, so the split misses an integer and the proof is incomplete. 3. Requirement #2: Mutually Exclusive Cases Ideally your cases are mutually exclusive : no scenario belongs to two cases at once. Overlap doesn't make a proof wrong , but it creates redundant work and room for confusion — so clean splits are preferred. Case 1: x < 0 Case 2: x = 0 Case 3: x > 0 No number satisfies two cases.

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