Proof by Counterexample
Proof Technique
Let $X$ be the statement:
- $\forall x \in S: P \left({x}\right)$
(For all the elements $x$ of a given set $S$, the property $P$ holds.)
Such a statement may or may not be true.
An example of such a statement which is definitely not true is:
- All Englishmen are cowards.
For example, it is generally accepted that Lord Horatio Nelson (whatever other character flaws he may or may not have had) was most definitely not a coward.
Let $Y$ be the statement:
- $\exists y \in S: \neg P \left({y}\right)$
(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
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (next): Preface
"... Any example which in some respect stands opposite to the reals is truly a Gegenbeispiel."
