ordered pair

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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 ${\displaystyle \langle a,b\rangle }$.

   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.

• triple, triplet
• tuple