Cartesian product

In mathematics, the Cartesian product (or direct product) of two sets X and Y, 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:

$X\times Y=\{(x,y)|x\in X\;\mathrm {and} \;y\in Y\}.$

The Cartesian product is named after René Descartes whose formulation of analytic geometry gave rise to this concept.

For example, if set X is the 13-element set { A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2 } and set Y is the 4-element set {♠, ♥, ♦, ♣}, then the Cartesian product of those two sets is the 52-element set { (A, ♠), (K, ♠), ..., (2, ♠), (A, ♥), ..., (3, ♣), (2, ♣) }.

Cartesian square and n-ary product

The Cartesian square (or binary Cartesian product) of a set X is the Cartesian product X × X. An example is the 2-dimensional plane 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).

This can be generalized to the n-ary Cartesian product over n sets X₁, ..., X_n:

$X_{1}\times \ldots \times X_{n}=\{(x_{1},\ldots ,x_{n})|x_{1}\in X_{1}\;\mathrm {and} \;\ldots \;\mathrm {and} \;x_{n}\in X_{n}\}.$

Indeed, it can be identified to (X₁ × ... × X_n-1) × X_n. It is a set of n-tuples.

An example of this is the Euclidean 3-space R × R × R, with R again the set of real numbers.

As an aid to its calculation, a table can be drawn up, 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.

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

$\prod _{i\in I}X_{i}=\{f:I\to \bigcup _{i\in I}X_{i}\ |\ (\forall i)(f(i)\in X_{i})\},$

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 i in I, the function

$\pi _{i_{0}}:\prod _{i\in I}X_{i}\to X_{i_{0}}$

defined by

$\pi _{i_{0}}(f)=f(i_{0}),\,$

is called the ith projection map.

An n-tuple can be viewed as a function on {1, 2, ..., n} that takes its value at i to be the ith 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 ith term in its corresponding set X_i. Once again, trusty old $\mathbb {R}$ provides an example of this:

$\prod _{n=1}^{\infty }\mathbb {R} =\mathbb {R} ^{\omega }=\mathbb {R} \times \mathbb {R} \times \ldots$

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 arbitrary non-empty collection of non-empty sets is non-empty is equivalent to the axiom of choice.

Cartesian product of functions

If f is a function from A to B and g is a function from X to Y, their cartesian product f×g is a function from A×X to B×Y with

$(f\times g)(a,x)=(f(a),g(x))$

As above this can be extended to tuples and infinite collections of functions.

Category theory

Categorically, the cartesian product is the product in the Category of sets.

See also

• Binary relation
• Direct product
• Empty product
• Product (category theory)
• Product topology
• Relation (mathematics)