Ordered pair

In mathematics, an ordered pair is a pair of elements in which order is significant: that is, the pair (x,y) is to be distinguished from (y,x). The ordered pairs (a,b) and (c,d) are equal if and only if a=c and b=d.

It would be possible to take the concept of ordered pair as an elementary concept in set theory, but it is more usual to define them in terms of sets. Kuratowksi proposed the definition

$(a,b)=\{\{a\},\{a,b\}\}.\,$

The set of all ordered pairs (x,y) with x in X and y in Y is the Cartesian product of X and Y. A complex number may be expressed as an ordered pair of real numbers, the real and imaginary parts respectively.

References

• Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 9-10. ISBN 0-387-90441-7. 
• Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold, 22-25. 
• Ian Stewart; David Tall (1977). The Foundations of Mathematics. Oxford University Press, 62-65. ISBN 0-19-853165-6.