The sentence:
He is a college student.
is not a sentence because it may be true or false depending on the value of the pronoun (variable) "he."
In the sentence:
Adam is a student at Kent State.
The predicate is the part of the sentence form which the subject has been removed. For example, if we remove the subject, Adam, from the sentence, Adam is a student at Kent State, then the remaining part, is a student at Kent State, is the predicate. That is,
In the sentence:
Adam is a Student at Kent State.
Noun = Adam
Predicate = is a student at Kent State
In logic, predicate can be obtained by removing any nouns from a statement. For example, consider the sentence again,
Adam is a student at Kent State.
Let
P = "is a student a Kent State"
and
Q = "is a student at"
Then P and Q are predicate symbols.
The sentence:
x is a student at Kent State.
is symbolized as P(x), where x is a predicate variable i.e.,
P(x) = x is a student at Kent State
And the sentence:
x is a student at y
is symbolized a Q(x, y), where x and y are predicate variables i.e.,
Q(x, y) = x is a student at y
Notice, when concrete values (from the appropriate set called domain) are substituted in place of predicate variables, a statement is result.
Formally,
| A predicate is a sentence that contains a
finite number of variables and becomes a statement when specific values
are substituted for the variables.
The domain of a predicate variable is the set of all values that may be substituted in place of the variables. |
Example of some domains are:
| R | = | Set of all real numbers. |
| Z | = | Set of all integers. |
| Q | = | Set of all rational numbers. |
When an element in the domain of the variable of a one-variable predicate is substituted for a variable, the resulting statement is either true or false. The set of all such elements that make the predicte true is called the truth set of the predicate.
Formally,
| If P(x) is a predicate and x has 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. 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)." |
As an example, let P(x) be "x is a factor of 8." Also, let the domain of x is the set of all positive integers, i.e., x ∈ Z+. Then, the truth set P(x) is {1, 2, 4, 8}. Since, 1, 2, 4, and 8 are exactly the positive integers that divide 8 exactly.
Another example, lets find the truth set of the predicate x > 1/x and suppose x ∈ R.
Using the definition of truth set {x ∈ R : x > 1/x} = {x ∈ R : x > 1 or -1 < x < 0}.
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