# Philosophy:Extension (predicate logic)

__: Set of tuples in mathematical logic that satisfy a predicate__

**Short description**The **extension** of a predicate – a truth-valued function – is the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation.

## Examples

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 [math]\displaystyle{ \Phi }[/math] can be written as

- [math]\displaystyle{ \{ (x_1,...,x_n) \mid \Phi(x_1,...,x_n) \}\,. }[/math]

## 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, respectively – making the characteristic function a predicate – , then for all relations *R* and predicates [math]\displaystyle{ \Phi }[/math] the following two statements are equivalent:

- [math]\displaystyle{ \Phi }[/math] is the characteristic function of
*R* *R*is the extension of [math]\displaystyle{ \Phi }[/math]

## See also

- Extensional logic
- Extensional set
- Extensionality
- Intension

## References

Original source: https://en.wikipedia.org/wiki/Extension (predicate logic).
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