Counterexamples

01A counterexample is a member of the stated domain that fails a universal claim#

A counterexample is a member of the stated domain that fails a universal claim. Consider: “Every label in $D=\{2,4,7\}$ is below $6$.” The symbol $x<6$ means that $x$ is below $6$.

Claim: every label in $D=\{2,4,7\}$ is below $6$. The labels $2$ and $4$ pass. The label $7$ belongs to $D$, but fails. That single failure refutes “every”.

For the claim “Every label in $D=\{2,4,7\}$ is below $6$,” an outside value is not a counterexample. The value $9$ fails the property, but is absent from $D$. The claim never covered it.

02Confirming examples do not cancel a failure#

Confirming examples do not cancel a failure. Even if many listed items pass a check, one listed failure disproves “Every listed item passes.” Finding a counterexample settles falsity; failing to find one in a partial search does not prove truth.

03Check your understanding#