ordered pair

English

ordered pair (plural ordered pairs)

(set theory) An object containing exactly two elements in a fixed order, so that, when the elements are different, exchanging them gives a different object. Notation: (a, b) or ${\\displaystyle \\langle a,b\\rangle }$ .
If an ordered pair were defined (in terms of sets) as ${\\displaystyle (x,y):=\\{\\{a\\},\\{a,\\{b\\}\\}\\}}$ then the "first element" of an ordered pair S could be defined as CAR(S) where CAR(S) = x if and only if ${\\displaystyle (\\forall y\\in S.\\,x\\in y)}$ . Likewise, the "second element" of S could be defined as CDR(S) where CDR(S) = x if and only if ${\\displaystyle (\\exists y\\in S.\\,(\\exists z\\in y.\\,x\\in z))}$ . If the two elements happened to be equal, then the ordered pair would still have cardinality two as would be naturally expected.

in set theory