Page 9 - Chapter 2

• Propositional function (predicate)

An element of a set {x | P(x)} is an object t for which the

statement P(t) is true. P(x) is called a propositional function

(or predicate) , because each choice of x produces a

proposition P(x) that is either true or false (well-defined)

E.g. Let A={ x | x is an integer less than 8}.

Here P(x) is the sentence “x is an integer less than 8”

P(1) denotes the statement “1 is an integer less than 8” (true)

P(8) denotes the statement “8 is an integer less than 8” (false)

• Universal Quantifiers (∀)

The Universal Quantifiers of a predicate P(x) is the statement

“for all values of x, P(x) is true” , denoted by ∀ x P(x)

Example 8

(a) P(x) : -(-x) = x is a predicate that makes sense for all real

number x.

then ∀ x P(x) is true statement. Since ∀ x ∈ R, -(-x) = x

(b) Q(x): x+1<4.

then ∀ x Q(x) is a false statement, since Q(5) is false