Disproof by Counterexample

Disproof by counterexample is one of the simplest yet most important techniques in mathematical reasoning. To disprove a universal statement — one that claims something is true for all values — you need only find one single example where the statement fails. This single example is called a counterexample.

The AQA A-Level Mathematics specification (7357) requires students to understand disproof by counterexample as one of the four key proof techniques. Questions on this topic appear regularly in exam papers and are typically accessible marks if you understand the technique.

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The Logic Behind Counterexamples

A universal statement has the form "For all x in S, property P(x) holds." In mathematical notation: ∀x ∈ S, P(x).

To disprove this, you need to find at least one x₀ ∈ S such that P(x₀) is false. In notation: ∃x₀ ∈ S such that ¬P(x₀).

One counterexample is sufficient to disprove a universal statement. It does not matter if the statement holds for millions of other values — a single failure is enough.

Conversely, no number of confirming examples constitutes a proof of a universal statement (unless you check every case, which is proof by exhaustion).

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