Extension (predicate logic)

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.

ExamplesEdit

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 functionEdit

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   the following two statements are equivalent:

  •   is the characteristic function of R
  • R is the extension of  

See alsoEdit

ReferencesEdit