1.4d Proof by Cases

Proof by Cases (Exhaustion)

A proof by cases is a logical argument that establishes the truth of a given proposition $P$ by dividing the problem into a finite number of exhaustive and mutually exclusive cases $P_1,\dots ,P_k$ and demonstrating that the proposition holds in at least one of the cases, such that
$(P_1\vee P_2\vee \dots \vee P_k)⟹P.$
Each proposition can be proven using any of the proof methods mentioned in this section.

Examples

There exist some irrational numbers $a,b\in \mathbb{R}\backslash \mathbb{Q}$, such that $x^y$ is rational.

We proceed by cases. Since the theorem