Quantifier/Examples/Existence of Multiplicative Identity

Example of Use of Quantifiers

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

$\exists x: \forall y: \exists z: \paren {y > z} \implies y = x z$

means:

There exists a natural number $x$ such that every natural number $y$ equals the product of $x$ with a natural number $z$.

This is shown to be true by setting $x = 1$.

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 $(7) \ \text{(iv)}$