ordered pair

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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.

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