Quantifier/Examples/Existence of x^y = y^x

Example of Use of Quantifiers

Let $x$ and $y$ be in the natural numbers.


$\exists x: \exists y: \paren {x \ne y} \land x^y = y^z$

means:

There exist distinct natural numbers $x$ and $y$ such that $x^y$ equals $y^x$.


This is true:

$2^4 = 16 = 4^2$


Sources

  • 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(7) \ \text{(iii)}$
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