Proof by Exhaustion and Counter Examples

Proof by exhaustion is a mathematical proof method in which every possible case in a finite set is considered.

Proof by exhaustion is also sometimes called proof by cases when all possible cases are examined separately.

For proof by exhaustion to be valid, the cases considered must cover every possible situation.

Proof by exhaustion is most useful when there is a small or manageable number of possible cases.

Each case in a proof by exhaustion must be checked using valid mathematical reasoning.

If the required statement is true in every possible case, the statement has been proved.

Checking only some examples is not proof by exhaustion because all possible cases must be considered.

A counterexample is a specific example that shows that a mathematical statement is false.

One valid counterexample is sufficient to disprove a universal statement.

A universal statement is a statement claiming that something is true for every value or every object in a specified set.

Counterexamples are particularly useful for statements containing words such as all, every, always or for any.

A counterexample must satisfy the conditions of the original statement while showing that its conclusion is false.

Finding several examples that support a statement does not prove that the statement is universally true.

Proof by exhaustion can establish that a finite collection of cases all satisfy a statement.

A counterexample has the opposite purpose: it demonstrates that a universal claim does not always hold.

When using proof by exhaustion, cases should be organised systematically so that none are accidentally omitted.

Proof by exhaustion may involve testing possible integer values, algebraic cases or geometric configurations.

Parity can be used to divide an exhaustion proof into cases such as even and odd integers.

A mathematical claim may sometimes be disproved more efficiently with a counterexample than by attempting a full proof.

Before proving a statement, it can be useful to test simple values to identify whether a counterexample exists.

If a counterexample is found, there is no need to continue trying to prove the original universal statement.

Proof by exhaustion is different from simply testing random numerical examples because exhaustion checks every allowable case.

A counterexample does not need to explain why a statement fails generally; it only needs to demonstrate one valid failure.

Good mathematical reasoning requires identifying whether a question asks you to prove a statement or disprove it.

In an examination, a proof by exhaustion should clearly identify all cases, demonstrate the result for each case and state the final conclusion.

When giving a counterexample, students should clearly show that the example meets the original conditions but contradicts the proposed conclusion.