Definition:Quantifier

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Definition

The universal quantifier $\forall$ and the existential quantifier $\exists$ are referred to collectively as quantifiers.


The Universal Quantifier

The symbol $\forall$ is called the universal quantifier.

It expresses the fact that, in a particular universe of discourse, all objects have a particular property.


That is:

$\forall x:$

means:

For all objects $x$, it is true that ...


In the language of set theory, this can be formally defined:

$\forall x \in S: \map P x := \set {x \in S: \map P x} = S$

where $S$ is some set and $\map P x$ is a propositional function on $S$.


The Existential Quantifier

The symbol $\exists$ is called the existential quantifier.

It expresses the fact that, in a particular universe of discourse, there exists (at least one) object having a particular property.


That is:

$\exists x:$

means:

There exists at least one object $x$ such that ...


In the language of set theory, this can be formally defined:

$\exists x \in S: \map P x := \set {x \in S: \map P x} \ne \O$

where $S$ is some set and $\map P x$ is a propositional function on $S$.


Also defined as

Some sources explicitly categorize the quantifiers as operators.


Examples

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 $x + y = 0$.


Also see

  • Results about quantifiers can be found here.


Sources

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