A counterexample to a proposition is an example that satisfies all its hypotheses but makes its conclusion
false. For a proposition asserting that 
is true for every 
, a counterexample is an element 
for which 
is false. Equivalently, a counterexample to 
must satisfy 
and not 
. Finding one counterexample therefore provides a proof
that a claim containing a universal quantifier
is false (Velleman 2019).
Conjectures are often based on extensive numerical or structural evidence. Confirming examples can strengthen the evidence for a conjecture but cannot prove a universal claim, whereas a single verified counterexample refutes it. A purported counterexample that violates one of the conjecture's hypotheses does not refute the conjecture. Counterexamples can also suggest improved conjectures by exposing a missing hypothesis or an overly strong conclusion (Lakatos 2015).
In a setting governed by the well ordering principle, one can also prove a universal claim by assuming a counterexample exists and choosing a least one. If the least counterexample yields a smaller counterexample, the resulting contradiction proves that no counterexample exists. This is a proof by contradiction.
Systematic collections of counterexamples are especially common in analysis and topology (Steen and Seebach 1995, Gelbaum and Olmsted 2003).
