gen-briefly-explain-proof-by-cases-and-one-common-misconception-5d7d

Briefly explain proof by cases and one common misconception about it.

# Proof by Cases **Proof by cases** (also called proof by exhaustion or case analysis) is a technique where you prove a statement by dividing all possibilities into separate cases and proving the statement holds for each case individually. **Structure:** 1. Show that your cases cover all possibilities 2. Prove the statement for each case separately 3. Conclude the statement is true overall **Simple example:** Prove that n² + n is always even for any integer n. - *Case 1:* n is even → n = 2k, so n² + n = 2k² + 2k = 2(k² + k) ✓ even - *Case 2:* n is odd → n = 2k + 1, so n² + n = (2k+1)² + (2k+1) = 4k² + 6k + 2 = 2(2k² + 3k + 1) ✓ even ## Common Misconception **Misconception:** You only need to prove *one* case or show an example from each case. **Reality:** You must prove the statement holds for **every** element in **every** case. Showing one example per case isn't sufficient—you need a complete argument that covers all instances within each case. Additionally, all cases together must be exhaustive (cover every possibility) and may overlap, which is fine as long as everything is covered.