Counterexample
Calculus & Advanced MathA 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.
Example
Statement: "All prime numbers are odd." Counterexample: $2$ is prime and even. One example is enough to disprove the whole statement.
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.
Definition
A counterexample to the statement "For all $x$, $P(x)$" is a specific value $x_0$ such that $P(x_0)$ is false. One counterexample completely disproves a universal claim. Searching for counterexamples is often the first step when investigating whether a conjecture is true.
Example
Conjecture: "$f'(x) = 0$ at a critical point implies a local extremum." Counterexample: $f(x) = x^3$ at $x = 0$: $f'(0) = 0$ but $x = 0$ is neither a max nor a min (it is an inflection point).
Key Insight
Before trying to prove something, always try to find a counterexample first. If you find one, you are done disproving. If you cannot, the search gives you intuition for why it might be true.
Definition
A counterexample witnesses the falsity of a universal statement. In model theory, a structure that satisfies the negation of a sentence is a counterexample model. Counterexample construction is a major research technique: the Weierstrass function (continuous but nowhere differentiable) was a counterexample to the intuition that continuous functions are almost everywhere differentiable.
Example
Banach-Tarski paradox: a counterexample to the intuition that rigid motions preserve volume. Using the Axiom of Choice, a ball can be decomposed and reassembled into two copies of itself.
Key Insight
Pathological counterexamples reveal the limits of intuition and motivate the precise conditions in theorems. Every "hypothesis" in a theorem statement typically exists because some counterexample shows it cannot be removed.