Ordered pairs
A stumbling block in the adoption of sets during the late nineteenth century was how to define an ordered pair, since the defintion of a set is not affected by the order of its elements.
That is to say that these two sets are the same:
\(\{a, b\}\) and \(\{b, a\}\)
In 1914 definitions were put forward by Norbert Weiner (1894-1964) and Felix Hausdorff (1868-1942), but both definitions were awkward.
In 1921 the Polish mathematician Kazimierz Kuratowsky (1896-1980) published the following definition, which has since become standard:
\(\{\{a\}, \{a, b\}\}\)
But, since this is rather cumbersome, the usual notation for ordered pairs would refer to this simply as \((a, b)\).
Ordered pairs are equal if the first elements of each set are equal and the second elements of each set are equal, in other words:
\((a, b) = (c, d)\) only if \(a = c\) and \(b = d\)
Ordered n-tuples
Generalised ordered pair notation denoting a set with any finite number of elements, taking order and multiplicity into account.
The ordered n-tuple \((x_1,x_2...,x_n)\) consists of the elements \(x_1, x_2..., x_3\) in the order written.
An ordered n-tuple of two elements is called an ordered pair.
An ordered n-tuple of three elements is called an ordered triple.
Ordered n-tuples are equal only if all elements of each set are equal and in the same order.