Editing Extensionality (section)

==In mathematics==
The extensional definition of function equality, discussed above, is commonly used in mathematics.
A similar extensional definition is usually employed for [[relation (mathematics)|relations]]: two relations are said to be equal if they have the same [[Extension (predicate logic)|extensions]].

In [[set theory]], the [[axiom of extensionality]] states that two [[set (mathematics)|sets]] are equal if and only if they contain the same elements.  In mathematics formalized in set theory, it is common to identify relations&mdash;and, most importantly, [[function (mathematics)|functions]]&mdash;with their extension as stated above, so that it is impossible for two relations or functions with the same extension to be distinguished.

Other mathematical objects are also constructed in such a way that the intuitive notion of "equality" agrees with set-level extensional equality; thus, equal [[ordered pair]]s have equal elements, and elements of a set which are related by an [[equivalence relation]] belong to the same [[equivalence class]].

[[Type theory|Type-theoretical]] foundations of mathematics are generally ''not'' extensional in this sense, and [[setoid]]s are commonly used to maintain a difference between intensional equality and a more general equivalence relation (which generally has poor [[constructivism (mathematics)|constructibility]] or [[Decidability (logic)|decidability]] properties).