Quantifier/Examples
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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):
$\epsilon$-$\delta$ 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:
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 $x = 1$.