# Unordered pair

In mathematics, an **unordered pair** or **pair set** is a set of the form {*a*, *b*}, i.e. a set having two elements *a* and *b* with **no particular relation between them** , where {*a*, *b*} = {*b*, *a*}. In contrast, an ordered pair (*a*, *b*) has *a* as its first element and *b* as its second element, which means (*a*, *b*) ≠ (*b*, *a*).

While the two elements of an ordered pair (*a*, *b*) need not be distinct, modern authors only call {*a*, *b*} an unordered pair if *a* ≠ *b*.^{[1]}^{[2]}^{[3]}^{[4]}
But for a few authors a singleton is also considered an unordered pair, although today, most would say that {*a*, *a*} is a multiset. It is typical to use the term unordered pair even in the situation where the elements a and b could be equal, as long as this equality has not yet been established.

A set with precisely two elements is also called a 2-set or (rarely) a **binary set**.

An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1.

In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing.

More generally, an **unordered ***n***-tuple** is a set of the form {*a*_{1}, *a*_{2},... *a _{n}*}.

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## Notes

[edit]**^**Düntsch, Ivo; Gediga, Günther (2000),*Sets, Relations, Functions*, Primers Series, Methodos, ISBN 978-1-903280-00-3.**^**Fraenkel, Adolf (1928),*Einleitung in die Mengenlehre*, Berlin, New York: Springer-Verlag**^**Roitman, Judith (1990),*Introduction to modern set theory*, New York: John Wiley & Sons, ISBN 978-0-471-63519-2.**^**Schimmerling, Ernest (2008),*Undergraduate set theory***^**Hrbacek, Karel; Jech, Thomas (1999),*Introduction to set theory*(3rd ed.), New York: Dekker, ISBN 978-0-8247-7915-3.**^**Rubin, Jean E. (1967),*Set theory for the mathematician*, Holden-Day**^**Takeuti, Gaisi; Zaring, Wilson M. (1971),*Introduction to axiomatic set theory*, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag

## References

[edit]- Enderton, Herbert (1977),
*Elements of set theory*, Boston, MA: Academic Press, ISBN 978-0-12-238440-0.