Quantifier/Examples

Examples of Use of Quantifiers

Existence for All of Element Greater Than

$\forall x: \exists y: x < y$

means:

For every $x$ there exists a $y$ such that $x < y$

or (assuming the domain is that of numbers):

For every number there exists a number greater than it.

$\epsilon$-$\delta$ Condition

$\forall \epsilon: \exists \delta: \forall y: \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon$

means:

For every $\epsilon$ there exists a $\delta$ such that for every $y$:

If $\size {x - y} < \delta$ then $\size {\map f x - \map f y} < \epsilon$.

Uniqueness of Additive Identity

$\forall x: \exists ! y: x + y = 0$

means:

For every $x$ there exists a unique $y$ such that $x + y = 0$.

Square of Sum

$\forall x: x \in \R \implies \paren {x + 2}^2 = x^2 + 4 x + 4$

means:

Every real number $x$ satisfies the equation $\paren {x + 2}^2 = x^2 + 4 x + 4$.

Equation involving Square

$\exists x: x \in \Z: x^2 + 2 = 11$

means:

There exists an integer $x$ which satisfies the equation $x^2 + 2 = 11$.

In the following, $x$, $y$ and $z$ are assumed to be in the domain of the natural numbers.

Existence for All of Twice Element

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

means:

Every natural number is twice a natural number.

This is false.

Thus:

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

Definition of Greater Than or Equal To

$\forall x: \forall y: \exists z: y \ge x \implies y = x + z$

means:

If a natural number $y$ is not less than a natural number $x$, then $y - x$ is a natural number.

Existence of $x, y \in \N$ such that $x^y = y^x$

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

means:

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

This is true:

$2^4 = 16 = 4^2$

Existence of Multiplicative Identity

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

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 $x = 1$.