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The extension of a predicatea truth-valued functionis the set of
tuple In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...
s of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation.


Examples

For example, the statement "''d2'' follows the weekday ''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 follows the weekday Saturday" (for example) is false. Using set-builder notation, the extension of the ''n''-ary predicate \Phi 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 \Phi the following two statements are equivalent: *\Phi is the characteristic function of ''R'' *''R'' is the extension of \Phi


See also

* Extensional logic * Extensional set *
Extensionality In logic, extensionality, or extensional equality, refers to principles that judge objects to be equality (mathematics), equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned wi ...
* Intension


References


extension (semantics)
in nLab Predicate logic {{mathlogic-stub