The universal quantifier is a quantifier that describes a predicate as true for all items in its domain. The universal quantifier is expressed by the symbol "" which is read as "for all". Formally this can be defined as follows

Let be a predicate and the domain of . A universal statement is a statement of the form "". It is defined to be true if, and only if, is true for every in . It is defined to be false if, and only if, is false for at least one in .

A value for which the predicate the universal quantifier is applied to which is false is called a counterexample to the universal statement.

Example

Given the predicate over the domain (the set of all natural numbers). We can use the universal quantifier to describe this predicate for all possible inputs as follows:

This can be read as "for all natural numbers , "

In this example, for the statement to be true has to be true for every value of in the set of all natural number . This example is false because there exist two counterexamples (1 and 2) in which is not true.

Negation

The negation of a universal quantifier is an existential quantifier with the condition negated.

Formally:

Example

What is the negation of the statement "All mathematicians wear glasses"?

“There is at least one mathematician who does not wear glasses.”

Implicit Universal Quantification

Occasionally, a statement will not include the words "for all" or "every" within it. The only clue to indicate that a statement has a universal quantification is the usage of an indefinite article ("a" or "an"). When this happens it's called an implicit universal quantification.

Example

"If a number is an integer, then it is a rational number"

This is an example of an implicit universal quantification as it uses the indefinite article "a".

Notation

Typically and are used to denote implicit universal quantification between two predicates. Formally given two predicates where there is a common domain of the notation means that every element in the truth set of is in the truth set of . Additionally, the notation means that and have identical truth sets.