Proof by Counterexample
Proof Technique
Consider the definition of a counterexample:
Let $X$ be the statement:
- $\forall x \in S: P \left({x}\right)$
That is:
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
- 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