Extension (Predicate Logic)

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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 functionis, by convention, the set of all such tuples associated with thevalue *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 <math>Phi</math> 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* and predicates <math>Phi</math> the following two statements are equivalent:

## See also

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For example the statement "

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

- <math>\,.</math>

- <math>Phi</math> is the characteristic function of
*R*; *R*is the extension of <math>Phi</math>.

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