Proof by Counterexample

Proof Technique

Consider the definition of a counterexample:

Let $X$ be the universal statement:

$\forall x \in S: \map P x$

That is:

For all the elements $x$ of a given set $S$, the property $P$ holds.

Such a statement may or may not be true.

Let $Y$ be the existential statement:

$\exists y \in S: \neg \map P y$

That is:

There exists at least one element $y$ of the set $S$ such that the property $P$ does not hold.

It follows immediately by De Morgan's laws that if $Y$ is true, then $X$ must be false.

Such a statement $Y$ is referred to as a counterexample to $X$.

Proving, or disproving, a statement in the form of $X$ by establishing the truth or falsehood of a statement in the form of $Y$ is known as the technique of proof by counterexample.

Internationalization

Counterexample is translated:

In German:

Gegenbeispiel

Sources

• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.7$: Counterexamples
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 3$: Statements and conditions; quantifiers
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic