The extension of a predicatea truth-valued functionis the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation.
For example the statement "d2 is the weekday following d1"
can be seen as a truth function associating to each tuple (d2, d1)
the value true or false. The extension of this truth function
is, by convention, the set of all such tuples associated with the
value true, i.e.
By examining this extension we can conclude that "Tuesday is the weekday following Saturday" (for example) is false.
Using set-builder notation, the extension of the n-ary predicate can be written as
Relationship with characteristic function
If the values 0 and 1 in the range of a characteristic function
are identified with the values false and true, respectivelymaking the characteristic function a predicate, then for all relations R
the following two statements are equivalent:
- is the characteristic function of R;
- R is the extension of .