Quantifiers

A quantifier is a logical symbol that specifies how many elements of a domain satisfy a given property.

There are two main types of quantifiers in mathematical logic:

• Universal quantifier $\forall$ (

  $\forall$

  ), read as "for all".
• Existential quantifier $\exists$ (

  $\exists$

  ), read as "there exists".

The universal quantifier states that a property holds for all elements in a given set. It is usually used in the following format:

$\forall x \in S, p(x)$ (

$\forall x \in S, p(x)$

)

which means "for all $x$ in the set $S$, the property $p(x)$ it true." (Here, $p(x)$ is a proposition that uses $x$ in it.)