what is the difference between exhaustive proof and proof by cases?

Question

I am currently reading Kenneth Rosen's book on Discrete Mathematics.

There I came across these two kind of proof strategy.

1. Proof By Cases
2. 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 ∨· · · ∨ p_n) → q the tautology [(p₁ ∨ p₂ ∨ · · · ∨ p_n) → q] ↔ [(p₁ → q) ∧ (p₂ → q) ∧ · · · ∧ (p_n → 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, ... , p_n can be proved by proving each of the n conditional statements p_i → 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 ∨ · · · ∨ p_n instead of p as the hypothesis of the conditional statement, where p and p₁ ∨ p₂ ∨ · · · ∨ p_n 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.

Thank You.

Answer 1

Proof by cases tends to be about splitting your proposition into cases, at least some of which are general while an exhaustive proof looks at every case in a way that is not general. For example if one were to prove a property about natural numbers, it might be easier to say prove it for even numbers and then prove it for odd numbers separately. This would be proof by cases. On the other hand, if one were to prove a property about, say, all natural numbers less than 10 then one could just check the proposition for each of the natural numbers. There would be no generality and this would be an exhaustive proof.

Here are two examples, see if you can decide which method might apply to which.

1. The maximum diameter for a tree on 10 vertices where no vertex has degree 2 is 5
2. Any isometry of $\mathbb R^3$ can be represented as the product of up to three reflections.