Proof by cases

Contents

Introduction

To prove a mathematical fact by cases, we break down the possible scenarios into a finite number, and prove the desired conclusion in each of these cases.

Straightforward direct proof shows that

R x & R y => (x = 0 | y = 0 == x * y = 0

That proof used the fact that

R x & R y & (x = 0 | y = 0) => x * y = 0

which is a natural one to approach by cases. One case is where x = 0, and the other is where y = 0. We prove the desired conclusion in each of these scenarios, and then combine the two results into a single statement.

The logical basis

The core mathematical fact behind proof by cases is that

(A => C) & (B => C) == A | B => C

The idea is to prove A => C and B => C separately, combine them into a single statement (A => C) & (B => C), then to replace that single statement with A | B => C.

The proof

Using the proof builder

Proving: R x & R y & (x = 0 | y = 0) => x * y = 0).