Proof by Cases

Mathematical Logic · Axiom Academy

When a disjunction holds, prove your conclusion separately for each case. This systematic approach transforms complex proofs into manageable pieces. Proof by cases exploits the logical rule of disjunction elimination (∨-elimination). The pattern is simple but powerful: Why does this work? Because P ∨ Q means "at least one is true." If R follows from P alone, and R also follows from Q alone, then R must be true regardless of which disjunct actually holds. 2. Why This Works: ∨-Elimination The logical foundation is the ∨-elimination rule from natural deduction: This is a complete way to eliminate a disjunction: you consider all possibilities. Since exactly one of P or Q must be true (or both), and R follows in either scenario, R is guaranteed. Not all case splits are explicitly given. Often, you create them by partitioning the domain. Here are common strategies: 4. Ensuring Cases Are Exhaustive The most critical requirement: your cases must be exhaustive . This means they cover every possibility—nothing falls through the cracks. You must justify that your cases are exhaustive. For well-known partitions (even/odd, positive/negative/zero), a brief note suffices. For custom partitions, provide explicit justification. 5. Proof Structure with Multiple Cases Here's the standard template for organizing a proof by cases: Proof by cases is particularly powerful in these situations: Absolute values: |x| splits naturally into x ≥ 0 and x < 0

This is the written version of the interactive lesson above. See the full Mathematical Logic course.