CARTESIAN PRODUCT

In mathematics, the 'Cartesian product' is a direct product of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept.
Specifically, the Cartesian product of two sets ''X'' (for example the points on an x-axis) and ''Y'' (for example the points on a y-axis), denoted ''X'' × ''Y'', is the set of all possible ordered pairs whose first component is a member of ''X'' and whose second component is a member of ''Y'' (e.g. the whole of the x-y plane):
:

X	imes Y = {(x,y) | xin X;mathrm{and};yin Y}.

For example, the Cartesian product of the thirteen-element set of standard playing card ranks {Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2} and the four-element set of card suits {♠, ♥, ♦, ♣} is the 52-element set of playing cards {(Ace, ♠), (King, ♠), ..., (2, ♠), (Ace, ♥), ..., (3, ♣), (2, ♣)}. The Cartesian product has 52 elements because that is the product of 13 times 4.
A Cartesian product of two finite sets can be represented by a table, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table, by choosing the element of the set from the row and the column.

n-ary product

Cartesian square and Cartesian power

Infinite products

Abbreviated form

Cartesian product of functions

Category theory

n-ary product

The Cartesian product can be generalized to the '''n''-ary Cartesian product' over ''n'' sets ''X''₁, ..., ''X_n'':
:

X_1	imescdots	imes X_n = {(x_1, ldots, x_n) mid  x_1in X_1;mathrm{and};cdots;mathrm{and};x_nin X_n}.

Indeed, it can be identified to (''X''₁ × ... × ''X_n-1'') × ''X_n''. It is a set of ''n''-tuples.

Cartesian square and Cartesian power

The 'Cartesian square' (or 'binary Cartesian product') of a set ''X'' is the Cartesian product ''X''² = ''X'' × ''X''.
An example is the 2-dimensional plane 'R²' = 'R' × 'R' where 'R' is the set of real numbers - all points (''x'',''y'') where ''x'' and ''y'' are real numbers (see the Cartesian coordinate system).
The ''cartesian power'' of a set ''X'' can be defined as:
:''X''^''n'' = ''X'' × ''X'' × .. × ''X'' =

{(x_1, ldots, x_n)|  x_1in X;mathrm{and};ldots;mathrm{and};x_nin X}.

An example of this is 'R³' = 'R' × 'R' × 'R', with 'R' again the set of real numbers, and more generally 'R^''n'''.
See also:

★ Euclidean space

★

Infinite products

The above definition is usually all that's needed for the most common mathematical applications. However, it is possible to define the Cartesian product over an arbitrary (possibly infinite) collection of sets. If ''I'' is any index set, and
:

{X_i | i in I}

is a collection of sets indexed by ''I'', then we define
:

prod_{i in I} X_i = { f : I 	o igcup_{i in I} X_i | (orall i)(f(i) in X_i)},

that is, the set of all functions defined on the index set such that the value of the function at a particular index ''i'' is an element of ''X_i'' .
For each ''j'' in ''I'', the function
:

pi_{j} : prod_{i in I} X_i 	o X_{j},

defined by
:

pi_{j}(f) = f(j),,

is called the '''j'' ^th projection map'.
An ''n''-tuple can be viewed as a function on {1, 2, ..., ''n''} that takes its value at ''i'' to be the ''i'' ^th element of the tuple. Hence, when ''I'' is {1, 2, ..., ''n''} this definition coincides with the definition for the finite case. In the infinite case this is a family.
One particular and familiar infinite case is when the index set is

mathbb N,

the natural numbers: this is just the set of all infinite sequences with the ''i'' ^th term in its corresponding set ''X_i''. Once again,

mathbb R

provides an example of this:
:

prod_{n = 1}^infty mathbb R =mathbb{R}^omega= mathbb R 	imes mathbb R 	imes cdots

is the collection of infinite sequences of real numbers, and it is easily visualized as a vector or tuple with an infinite number of components. Another special case (the above example also satisfies this) is when all the factors ''X_i'' involved in the product are the same, being like "Cartesian exponentiation." Then the big union in the definition is just the set itself, and the other condition is trivially satisfied, so this is just the set of ''all'' functions from ''I'' to ''X.''
Otherwise, the infinite cartesian product is less intuitive; though valuable in its applications to higher mathematics.
The assertion that the Cartesian product of an arbitrary collection of non-empty sets is non-empty is equivalent to the axiom of choice.

Abbreviated form

If several sets are being multiplied together, e.g.

X_1, X_2, X_3, ...

, then some authors ^[1] choose to abbreviate the Cartesian product as simply

imes X_i