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Predicates and Quantified statements

A predicate is a sentence that contains a finite

number of variables and becomes a statement

when specific values are substituted for the

variables.

. – p.1/23

Predicates and Quantified statements

For example, the sentence "x is a student of y" is

a predicate. We denote this as P (x, y). x and y

are called predicate variables that take values in

appropriate sets.

. – p.1/23

Predicates and Quantified statements

For example, the sentence "x is a student of y" is

a predicate. We denote this as P (x, y). x and y

are called predicate variables that take values in

appropriate sets.

The domain of a predicate variable is the set of

all values that may be substituted in place of the

variables (E.g.: The domain for x may be students

in NUS).

. – p.1/23

Truth Set

If P (x) is a predicate and x has the domain D,

the truth set of P (x) is the set of all elements of D

that make P (x) true when substituted for x.

. – p.2/23

Truth Set

The truth set of P (x) is denoted

{x ∈ D|P (x)}

which is read ”the set of all x in D such that P (x)”.

. – p.2/23

Example

Let D be the set of integers. Let P (x) be ”x is a

factor of 6”. Then the truth set

{x ∈ D|P (x)} = {1, 2, 3, 6}.

. – p.3/23

The use of ⇒ and ⇔

Let P (x) and Q(x) be predicates and suppose

the common domain of x is D.

. – p.4/23

The use of ⇒ and ⇔

Let P (x) and Q(x) be predicates and suppose

the common domain of x is D.

P (x) ⇒ Q(x) means that every element in the

truth set of P (x) is in the truth set of Q(x).

. – p.4/23

The use of ⇒ and ⇔

Let P (x) and Q(x) be predicates and suppose

the common domain of x is D.

P (x) ⇔ Q(x) means that the truth sets of P (x)

and Q(x) are identical.

. – p.4/23

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