predicate and quantifires (PDF)

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