# Quantifier

## Quantifier

Symbol indicating that a property is applied to all of the elements in a set or only to some of them.

### Symbols

• The universal quantifier is “$$\forall$$x ∈ E ” which is read as: “for any x element of set E” or “given any x element of set E”.
• The existential quantifier is “$$\exists$$x ∈ E” which is read as: “there is at least one x element of set E”.
• The uniqueness quantifier is $$\underset{1}{\exists}$$ which is read as: “there is one and only one x element of set E”.

### Examples

• The expression “$$\forall$$x ∈ $$\mathbb{N}$$ : (x + 4) ∈ $$\mathbb{N}$$” is read as: “given any whole number x, (x + 4) is a whole number”.
• The expression “$$\exists$$x ∈ $$\mathbb{N}$$ : x is a divisor of 12″ is read as: “there is at least one whole number x for which x is a divisor of 12″.
• The expression “$$\forall a, b ∈ \mathbb{R},\space \textrm{with }b ≠ 0, \underset{1}{\exists}x ∈ \mathbb{R} : \dfrac{a}{b} = x$$, is read as “for all numbers a and b of the set of real numbers, b being zero, there is one and only one x element of the set of real numbers for which the quotient of a by b is equal to x“.