Definition
Counterexample
A counterexample is one case that fits a claim's conditions but makes its conclusion false.
What it means
When a claim says something works for every number or every shape, you only need one valid rebel case to show the claim is wrong. The case must belong to the group named in the claim; otherwise it does not count.
Remember it
Every claim meets one honest exception: one valid case can topple an ‘every’ claim.
Example
Claim: ‘Every multiple of 4 ends in 4.’ The number 8 is a multiple of 4, but it ends in 8, so 8 is a counterexample and the claim is false.
Watch for
A case outside the claim's conditions is not a counterexample. For ‘all rectangles have four sides,’ a triangle cannot disprove the claim because it is not a rectangle.
Common mix-up
A counterexample disproves one universal claim; it does not automatically prove a new universal claim.
Exact meaning
A counterexample to a universal mathematical claim is a case that satisfies every stated condition of the claim but does not satisfy its conclusion. One counterexample is enough to disprove that claim.