ORDERED PAIR

ordered pair: translation

An object associated with a set of two objects considered in a particular order. Thus the ordered pair <a,b> is not identical with the ordered pair <b,a>, just as the queue ‘a first and b next’ is not the queue ‘b first and a next’. In set theory, ordered pairs cannot be identified with the set of their members, since a,b = b,a, which is exactly what we do not want. The task is to find sets with the property that <a,b> ? <b,a> if a ? b, and with the property that if <a,b> = <u,v> then a = u and b = v. One solution is that of Kazimierz Kuratowski (b. 1896), that identifies the ordered pair <a,b> with the set a,a,b. This is the definition in general use today. See also Cartesian product.