An [[Existence Proof]] is a type of [[Proof]] which establishes [[Proposition|Propositions]] of the form $\exists x\,P(x)$. An [[Existence Proof]] can either be [[Constructive Proof]] or [[Nonconstructive Proof]]. A constructive [[Proof]] finds an $a\in D$ such that $P(a)$ is [[Truth Value|true]], while a destructive [[Proof]] shows existence in any other way. >[!example] Non Constructive Proof >[[Proposition]]: There exists [[Irrational Numbers]] $x, y \in \Q'$ such that $x^{y} \in \Q$ > >If $\sqrt{ 2 }^\sqrt{ 2 }$ is [[Rational Numbers|Rational]] the [[Proposition]] is proved, $x=y=\sqrt{ 2 }$ > >If $\sqrt{ 2 }^\sqrt{ 2 }$ is not rational, then $\let x= \sqrt{ 2 }^\sqrt{2 }, y=\sqrt{ 2 }$. Then >$x^y=\pa{\sqrt{ 2 }^\sqrt{ 2 }}^\sqrt{ 2 }=\sqrt{ 2 }^{\sqrt{ 2 }\sqrt{ 2 }}=\sqrt{ 2 }^2=2$, which is rational. > >The [[Proposition]] is [[Truth Value|proved]].