I am currently reading Kenneth Rosen's book on Discrete Mathematics.
There I came across these two kind of proof strategy.
- Proof By Cases
- Exhaustive Proof
But I was not able to figure out the differences among them.
Here are the definitions mentioned in the book.
To prove a conditional statement of the form (p1 ∨ p2 ∨· · · ∨ pn) → q the tautology [(p1 ∨ p2 ∨ · · · ∨ pn) → q] ↔ [(p1 → q) ∧ (p2 → q) ∧ · · · ∧ (pn → q)] can be used as a rule of inference. This shows that the original conditional statement with a hypothesis made up of a disjunction of the propositions p1, p2, ... , pn can be proved by proving each of the n conditional statements pi → q, i = 1, 2, . . . , n, individually. Such an argument is called a proof by cases. Sometimes to prove that a conditional statement p → q is true, it is convenient to use a disjunction p1 ∨ p2 ∨ · · · ∨ pn instead of p as the hypothesis of the conditional statement, where p and p1 ∨ p2 ∨ · · · ∨ pn are equivalent.
EXHAUSTIVE PROOF : Some theorems can be proved by examining a relatively small number of examples. Such proofs are called exhaustive proofs, or proofs by exhaustion because these proofs proceed by exhausting all possibilities. An exhaustive proof is a special type of proof by cases where each case involves checking a single example.
Please provide an example along with the explanations to illustrate the differences.