Quantifier/Examples/Existence for All of Twice Element

Example of Use of Quantifiers

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

$\forall x: \exists y: x = y + y$

means:

Every natural number is twice a natural number.

This is false.

Thus:

$\exists x: \forall y: x \ne y + y$

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