Definition:Counterexample

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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 $X$.


Examples

Sum of Cubes of Digits

Let $P$ be the statement:

There exists no integer which is the sum of the cubes of its digits.


A counterexample to $P$ is the number $153$, as can be seen in Pluperfect Digital Invariants: $3$ Digits:

\(\ds 153\) \(=\) \(\ds 1 + 125 + 27\)
\(\ds \) \(=\) \(\ds 1^3 + 5^3 + 3^3\)


Sum of Sine and Cosine equals $1$

Let $P$ be the statement:

$\forall x \in \R: \cos x + \sin x = 1$


A counterexample to $P$ is the real number $\dfrac \pi 4 \in \R$:

\(\ds \cos \dfrac \pi 4\) \(=\) \(\ds \dfrac {\sqrt 2} 2\) Cosine of $\dfrac \pi 4$
\(\ds \sin \dfrac \pi 4\) \(=\) \(\ds \dfrac {\sqrt 2} 2\) Sine of $\dfrac \pi 4$
\(\ds \leadsto \ \ \) \(\ds \cos \dfrac \pi 4 + \sin \dfrac \pi 4\) \(=\) \(\ds \sqrt 2\) \(\ds \ne 1\)


Also see

  • Results about counterexamples can be found here.


Internationalization

Counterexample is translated:

In German: Gegenbeispiel


Sources

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