Counterexample — Definition, Meaning & Examples

Counterexample

An example which disproves a proposition. For example, the prime number 2 is a counterexample to the statement "All prime numbers are odd."

Example

Problem: Someone claims: "The square of any integer is greater than that integer." Find a counterexample to disprove this statement.

Step 1: The claim says that for every integer

n

, we have

n^2 > n

. To disprove it, we need just one integer where this fails.

Step 2: Try

n = 1

. Compute

n^2

and compare it to

n

1^2 = 1

Step 3: Since

1^2 = 1

, we have

1^2 = 1

, not

1^2 > 1

. The claim fails for this value.

Answer: The integer

n = 1

is a counterexample because

1^2 = 1

, which is not greater than

1

. This single example is enough to disprove the claim.

Why It Matters

Counterexamples are one of the most powerful tools in mathematical reasoning. While proving a statement true can require showing it holds for every possible case, disproving it requires only a single counterexample. This idea appears throughout algebra, geometry, and proof-writing courses whenever you need to test whether a conjecture is valid.

Common Mistakes

Mistake: Thinking that one supportive example proves a statement is always true.

Correction: A single example that satisfies a claim does not prove it. However, a single example that violates the claim does disprove it. Proof and disproof are not symmetric — disproving requires just one counterexample, but proving requires a general argument.

Related Terms

Prime Number — Often used in classic counterexample illustrations
Conjecture — An unproven claim that counterexamples can disprove
Proof — A rigorous argument that a statement is true
Theorem — A proven statement with no possible counterexample