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:

$\forall x \in S: P \left({x}\right) := \left\{{x \in S: P \left({x}\right)}\right\} = S$

where $S$ is some set and $P \left({x}\right)$ 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:

$\exists x \in S: P \left({x}\right) := \left\{{x \in S: P \left({x}\right)}\right\} \ne \varnothing$

where $S$ is some set and $P \left({x}\right)$ is a propositional function on $S$.


Also known as

Some sources categorize the quantifiers as operators.


Sources