Definition:Universal Quantifier

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

It expresses the fact that, in a particular universe of discourse, all object 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:
 * $$\forall x \in S: P \left({x}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{x \in S: P \left({x}\right)}\right\} = S$$

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

Some authors use $$\left({x}\right)$$ to mean $$\forall x$$, but the bespoke symbol is usually preferred as there is then no room for ambiguity.

Also see

 * Universal statement.