# Extension (predicate logic)

The **extension** of a predicateTemplate:Spaced ndasha truth-valued functionTemplate:Spaced ndashis 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.

{(Monday, Sunday), (Tuesday, Monday), (Wednesday, Tuesday), (Thursday, Wednesday), (Friday, Thursday), (Saturday, Friday), (Sunday, Saturday)}

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, respectivelyTemplate:Spaced ndashmaking the characteristic function a predicateTemplate:Spaced ndash, then for all relations *R* and predicates the following two statements are equivalent: