gen-define-proof-by-cases-and-give-one-short-worked-example-dfa2

Define proof by cases and give one short worked example.

# Proof by Cases ## Definition **Proof by cases** (also called *proof by exhaustion* or *case analysis*) is a method where you break a statement into several distinct cases that cover all possibilities, then prove the statement is true for each case separately. The logic: If you want to prove "P implies Q" and P can occur in ways Case 1, Case 2, ..., Case n (which cover all possibilities), then proving Q holds in each case proves Q always holds when P is true. ## Worked Example **Theorem:** For any integer n, n² + n is even. **Proof:** Any integer n is either even or odd. We consider both cases: **Case 1:** n is even - Then n = 2k for some integer k - n² + n = (2k)² + 2k = 4k² + 2k = 2(2k² + k) - Since this is 2 times an integer, n² + n is even ✓ **Case 2:** n is odd - Then n = 2k + 1 for some integer k - n² + n = (2k + 1)² + (2k + 1) = 4k² + 4k + 1 + 2k + 1 = 4k² + 6k + 2 = 2(2k² + 3k + 1) - Since this is 2 times an integer, n² + n is even ✓ Since n² + n is even in all possible cases, the theorem is proved. ∎