The universal quantifier is for all or every, and is represented by \(\forall.\) It is used to make statements about all elements of a set.

For example, let \(S = \{1, 2, 3, 4\}.\) We will represent the statement "every element of \(S\) is less than \(5\)" with predicate logic.

The predicate will be \(P(x)=\)"\(x\) is less than \(5\)".

The statement "every element of \(S\) is less than \(5\)" can be written
\[\forall x \in S, P(x)\]

The Existential Quantifier

The existential quantifier is there exists and is represented by \(\exists.\) It is used to make the claim that a statement is true about at least one elment of a set.

The existential quantifier symbol \(\exists\) is used with the such that symbol \(\ni.\) The statement \[\exists x \ni x < 2\] is read "there exists an \(x\) such that \(x\) is less than 2."

For example, let \(S\) be the set of all cars. We will write the statement "there are purple cars" with predicate logic.

The predicate will be \(P(x)=\)"\(x\) is a purple car."

The statement "there exists purple cars" can be written
\[\exists x \in S \ni P(x)\]

The following is true about the set \(S:\)
\[\forall x \in S, x > 5\]
Select all the elements that could be in \(S\):

The following is true about \(R:\)
\[\exists x \in R \ni x < 1\]
Which of the following sets could be \(R?\)