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Tuples, vectors, matrices/R/Sets/Introduction/Section

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Important product sets are and . The ordering of the elements is essential. In general, for a set and some , we denote the -th fold product set of with itself as

The elements have the form

where every is from . Such an ordered finite sequence of elements is also called an -tuple over . For , it is called a pair, for , it is called a triple. For

the element is called the -th component or the -th entry of the tuple. In this context, the is called the index of the tuple, and is called the index set of the tuple.

More generally, for every index set , there exist -tuples. In such an -tuple, to every index some mathematical object is assigned; the tuple is often written as , . If all are from one set , then we call this an -tuple from . For , we call this a sequence in .

A finite index set can always be replaced by a set of the form (this procedure is called a numbering of the index set), but this is not always useful. If we start with the index set

and if we are interested in a certain subset , then it is natural to stick to the original notation from instead of introducing a new numbering for . Quite often, there is a "natural“ index set for a certain problem that represents a part of the structure of the problem (and is easier to remember).

An -tuple over a set of the form

is also called a row tuple (of length ), and an -tuple of the form

is called a column tuple. These are just two different ways to represent the tuple, but if the tuple represents some structure (like a vector, to which a matrix (see below) shall be applied), then this difference is relevant.

When and are two sets and is their product set, then we can express an -tuple in as a "table“, that assigns to every pair an element . In particular, for and , we call an -tuple also an -matrix, and write this as

The row tuple

is called the -th row of the matrix, and

is called the -th column of the matrix.