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 "[latex]\forall[/latex]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 "[latex]\exists[/latex]x ∈ E" which is read as: "there is at least one x element of set E".
- The uniqueness quantifier is [latex]\underset{1}{\exists}[/latex] which is read as: "there is one and only one x element of set E".
Examples
- The expression "[latex]\forall[/latex]x ∈ [latex]\mathbb{N}[/latex] : (x + 4) ∈ [latex]\mathbb{N}[/latex]" is read as: "given any whole number x, (x + 4) is a whole number".
- The expression "[latex]\exists[/latex]x ∈ [latex]\mathbb{N}[/latex] : 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 "[latex]\forall a, b ∈ \mathbb{R},\space \textrm{with }b ≠ 0, \underset{1}{\exists}x ∈ \mathbb{R} : \dfrac{a}{b} = x[/latex], 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".