Definition:Counterexample

Definition

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 $X$.

Also see

• Proof by Counterexample

Internationalization

Counterexample is translated:

In German:

Gegenbeispiel

Sources

• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.7$: Counterexamples
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Preface

... Any example which in some respect stands opposite to the reals is truly a Gegenbeispiel.

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 3$: Statements and conditions; quantifiers
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): counterexample
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): counterexample
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): counterexample