gen-state-proof-by-cases-precisely-and-explain-why-it-matters-in-ee03

State proof by cases precisely and explain why it matters in 2-3 sentences.

## Proof by Cases **Precise Statement:** To prove a statement P, if we can partition all possibilities into exhaustive cases C₁, C₂, ..., Cₙ (where every scenario falls into exactly one case), and we prove P holds in each individual case, then P is true universally. Formally: If (C₁ ∨ C₂ ∨ ... ∨ Cₙ) is a tautology, and we prove (C₁ → P), (C₂ → P), ..., (Cₙ → P), then P is true. **Why it matters:** This technique is essential because many mathematical objects or situations don't behave uniformly—they have fundamentally different properties depending on context (e.g., positive vs. negative numbers, even vs. odd integers). By dividing complex problems into manageable subcases with distinct behaviors, we can apply case-specific reasoning that would be impossible or unnecessarily complicated in a single unified argument.