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.