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“.