Proof by Counterexample

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Proof Technique

Consider the definition of a counterexample:

Let $X$ be the statement:

$\forall x \in S: P \left({x}\right)$

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 statement:

$\exists y \in S: \neg P \left({y}\right)$

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