# ordered pair

## English

Wikipedia has an article on:

Wikipedia

### Noun

ordered pair (plural ordered pairs)

1. (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 $\langle a, b\rangle$.
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.