Counterexample
A counterexample is a specific case that disproves a general statement by showing it fails for at least one instance.
Definition
A counterexample is one specific example that proves a statement is false; you only need one case where the rule breaks down to disprove a "for all" claim. Formally, a counterexample to "For all $x$, $P(x)$" is a specific value $x_0$ such that $P(x_0)$ is false, and one counterexample completely disproves a universal claim. In model theory, a structure that satisfies the negation of a sentence is a counterexample model; counterexample construction is a major research technique, and every "hypothesis" in a theorem statement typically exists because some counterexample shows it cannot be removed.
Example
Statement: "All prime numbers are odd." Counterexample: $2$ is prime and even, one example is enough to disprove the whole statement. Consider the conjecture "$f'(x) = 0$ at a critical point implies a local extremum": the counterexample $f(x) = x^3$ at $x = 0$ has $f'(0) = 0$ but $x = 0$ is an inflection point, neither a max nor a min. The Weierstrass function (continuous but nowhere differentiable) was a counterexample to the intuition that continuous functions are almost everywhere differentiable, and the Banach-Tarski paradox, using the Axiom of Choice, is a counterexample to the intuition that rigid motions preserve volume.
Key Insight
In math, proving something true requires showing it works for ALL cases, but disproving it only requires finding ONE case where it fails, so always try to find a counterexample first before attempting a proof. Pathological counterexamples reveal the limits of intuition and motivate the precise conditions in theorems.