Editing Ordered pair (section)

====Variants====
The above Kuratowski definition of the ordered pair is "adequate" in that it satisfies the characteristic property that an ordered pair must satisfy, namely that <math>(a,b) = (x,y) \leftrightarrow (a=x) \land (b=y)</math>. In particular, it adequately expresses 'order', in that <math>(a,b) = (b,a)</math> is false unless <math>b = a</math>. There are other definitions, of similar or lesser complexity, that are equally  adequate:
* <math>( a, b )_{\text{reverse}} :=  \{ \{ b \}, \{a, b\}\};</math>
* <math>( a, b )_{\text{short}} :=  \{ a, \{a, b\}\};</math>
* <math>( a, b )_{\text{01}} := \{\{0, a \}, \{1, b \}\}.</math><ref>This differs from Hausdorff's definition in not requiring the two elements 0 and 1 to be distinct from ''a'' and ''b''.</ref>
The '''reverse''' definition is merely a trivial variant of the Kuratowski definition, and as such is of no independent interest. The definition '''short''' is so-called because it requires two rather than three pairs of [[braces (punctuation)|braces]]. Proving that '''short''' satisfies the characteristic property requires the [[Zermelo–Fraenkel set theory]] [[axiom of regularity]].<ref>Tourlakis, George (2003) ''Lectures in Logic and Set Theory. Vol. 2: Set Theory''. Cambridge Univ. Press. Proposition III.10.1.</ref> Moreover, if one uses [[Set-theoretic definition of natural numbers#Definition as von Neumann ordinals|von Neumann's set-theoretic construction of the natural numbers]], then 2 is defined as the set {0, 1} = {0, {0}}, which is indistinguishable from the pair (0, 0)<sub>short</sub>. Yet another disadvantage of the '''short''' pair is the fact that, even if ''a'' and ''b'' are of the same type, the elements of the '''short''' pair are not. (However, if ''a''&nbsp;=&nbsp;''b'' then the '''short''' version keeps having cardinality 2, which is something one might expect of any "pair", including any "ordered pair".)