Quantifiers

Quantifiers are words, expressions, or phrases that indicate the number of elements that a statement pertains to. In mathematical logic, there are two main quantifiers: the universal quantifier ("for all") and the existential quantifier ("there exists").

Universal quantifier: This quantifier is used to state a proposition that is true for all variables x of a given set. For example, the proposition "x^2 is a nonnegative number" is true for every real number, so we state "for all real numbers $x$, $x^{2}$ is nonnegative". Symbolically, we write the universal quantifier as $\forall$ (for example, $\forall x\in \mathbb {R} ,x^{2}\geq 0$).

Existential quantifier: This quantifier is used to state a proposition that is at least true for one element x of a given set. For example, the proposition "x is greater than 7" is not true for all real numbers, but we know it is true for some. So, we state "there exists a real number x such that x is greater than 7". Symbolically, we write the universal quantifier as $\exists$ (for example, $\exists x\in \mathbb {R} ,x>7$).

If there exists exactly one element such that a proposition is true, we can denote this statement with the symbol $\exists !$ (for example, $\exists !x\in \mathbb {R}$ such that $7x+1=0$, since there is exactly one solution to this equation).

If there exists no element such that a proposition is true, we can use the symbol $\nexists$ (for example, $\nexists x\in \mathbb {R}$ such that $2x=2x+1$).

Examples of how to use quantifiers:

• $\forall n\in \mathbb {N} ,n>0$
• $\exists a,b\in \mathbb {R} ,a+b=0$
• $\exists !x\in \mathbb {R} ,x^{2}=0$
• $\nexists z\in \mathbb {R} ,z={\sqrt {(-1)}}$

Resources

• Quantifiers, Mathematics LibreTexts