Ordered pair

English

Noun

1. Template:Lb 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 $(\mathrm{\forall }y\in S.x\in y)$. Likewise, the "second element" of S could be defined as CDR(S) where CDR(S) = x if and only if $(\mathrm{\exists }y\in S.(\mathrm{\exists }z\in y.x\in z))$. 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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