ordered pair

English

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 $⟨a,b⟩$.

   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.

Related terms

• triple, triplet
• tuple

Translations

in set theory

• Chinese:

  Mandarin: 有序對 / 有序对 (yǒuxùduì)

• Czech: uspořádaná dvojice f
• Finnish: järjestetty pari
• French: couple (fr) m
• Galician: par ordenado m
• German: geordnetes Paar (de) n
• Greek: διατεταγμένο ζεύγος (el) n (diatetagméno zévgos)
• Hungarian: rendezett pár (hu)
• Italian: coppia (it), coppia ordinata
• Japanese: 順序対 (じゅんじょつい, junjo-tsui)
• Korean: 순서쌍 (sunseossang)
• Māori: takirua raupapa
• Polish: para uporządkowana f
• Portuguese: par ordenado m
• Russian: упоря́доченная па́ра f (uporjádočennaja pára), па́ра (ru) f (pára)
• Spanish: par ordenado m
• Swedish: ordnat par
• Tagalog: kapid na ayos