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 $⟨ a, b ⟩$ .
If an ordered pair were defined (in terms of sets) as $(x, y) := {{a}, {a, {b}}}$ then the "first element" of an ordered pair S could be defined as CAR(S) where CAR(S) = x if and only if $(\forall y \in S . x \in y)$ . Likewise, the "second element" of S could be defined as CDR(S) where CDR(S) = x if and only if $(\exists y \in S . (\exists z \in y . x \in z))$ . 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