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Wikipedia / Cartesian product

In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation

A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form .

One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.

The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.

Set-theoretic definition

A rigorous definition of the Cartesian product requires a domain to be specified in the set-builder notation, with the typical Kuratowski's definition denotes the power set would be defined as

Examples

A deck of cards

An illustrative example is the standard 52-card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits } form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.

returns a set of the form {(A,♠), (A,), (A,), (A,♣), (K,♠), ..., (3,♣), (2,♠), (2,), (2,), (2,♣)}.

returns a set of the form {(♠,A), (♠,K), (♠,Q), (♠,J), (♠,10), ..., (♣,6), (♣,5), (♣,4), (♣,3), (♣,2)}.

These two sets are distinct, even disjoint, but there is a natural bijection between them, under which (3,♣) corresponds to (♣,3) and so on.

A two-dimensional coordinate system

The main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way, and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates. Usually, such a pair's first and second components are called its and denoting the real numbers) is thus assigned to the set of all points in the plane.

Most common implementation (set theory)

A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, Kuratowski's definition represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, and specification. Since functions are usually defined as a special case of relations, and relations are usually defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is necessarily prior to most other definitions.

Non-commutativity and non-associativity

Let , , and be sets.

The Cartesian product is not commutative because the ordered pairs are reversed unless at least one of the following conditions is satisfied:

is equal to , or
or is the empty set.

For example:

;

Strictly speaking, the Cartesian product is not associative If for example , then .

Intersections, unions, and subsets

The Cartesian product satisfies the following property with respect to intersections

In most cases, the above statement is not true if we replace intersection with union

Here are some rules demonstrating distributivity with other operators (see leftmost picture): denotes the absolute complement of .

Other properties related with subsets are:

Cardinality

The cardinality of a set is the number of elements of the set. For example, defining two sets: and . Both set and set consist of two elements each. Their Cartesian product, written as , results in a new set which has the following elements:

where each element of is paired with each element of , and where each pair makes up one element of the output set. The number of values in each element of the resulting set is equal to the number of sets whose Cartesian product is being taken; 2 in this case. The cardinality of the output set is equal to the product of the cardinalities of all the input sets. That is,

In this case,

Similarly,

and so on.

The set is infinite if either or is infinite, and the other set is not the empty set.

Cartesian products of several sets

n-ary Cartesian product

The Cartesian product can be generalized to the -ary Cartesian product over sets

of n-tupless. If tuples are defined as nested ordered pairs, it can be identified with . If a tuple is defined as a function on } that takes its value at to be the -th element of the tuple, then the Cartesian product

Cartesian nth power

The Cartesian square of a set is the Cartesian product . An example is the 2-dimensional plane where is the set of real numbers: is the set of all points where and are real numbers (see the Cartesian coordinate system).

The Cartesian th power of a set

An example of this is , with again the set of real numbers, and more generally .

The Cartesian th power of a set may be identified with the set of the functions mapping to the n-tuples of elements of . As a special case, the Cartesian 0th power of is the singleton set, that has the empty function with codomain as its unique element.

Intersections, unions, complements and subsets

. Then

;
;

, moreover, equality is possible only in the following cases:

;
.
can be calculated, if a universe. To simplify the expressions, we introduce the following notation. Let us denote the Cartesian product as a tuple bounded by square brackets; this tuple includes the sets from which the Cartesian product is formed, e.g.:
.

In n-tuple algebra (NTA), such a matrix-like representation of Cartesian products is called a C-n-tuple.

With this in mind, the union of some Cartesian products given in the same universe can be expressed as a matrix bounded by square brackets, in which the rows represent the Cartesian products involved in the union:

Such a structure is called a C-system in NTA.

will look like the following C:

In NTA, a diagonal C, that represents the complement of a C-n, can be written concisely as a tuple of diagonal components bounded by inverted square brackets:

This structure is called a D-n-tuple. Then the complement of the C. Such a structure is called a D-system and is calculated, if necessary, as the intersection of the D-n-tuples contained in it. For instance, if the following C-system is given:

then its complement will be the D-system

Let us consider some new relations for structures with Cartesian products obtained in the process of studying the properties of NTA. The structures defined in the same universe are called homotypic ones.

The intersection of C-systems. Assume the homotypic C. Their intersection will yield a C-system containing all non-empty intersections of each C-n with each C-n.
Checking the inclusion of a C-n-tuple into a D-n-tuple. For the C-n and the D-n.
Checking the inclusion of a C-n-tuple into a D-system. For the C-n and the D, if and only if, for every D-n.

Infinite Cartesian products

It is possible to define the Cartesian product of an arbitrary (possibly infinite) indexed family of sets. If is any index set is a family of sets indexed by that is, the set of all functions defined on the index set such that the value of the function at a particular index is an element of Xi. Even if each of the Xi is nonempty, the Cartesian product may be empty if the axiom of choice.

For each in is called the -th projection map.

Cartesian power is a Cartesian product where all the factors Xi are the same set is the set of all functions from to , and is frequently denoted XI. This case is important in the study of cardinal exponentiation, the natural numbers: this Cartesian product is the set of all infinite sequences with the -th term in its corresponding set Xi can be visualized as a vector.

Other forms

Abbreviated form

If several sets are being multiplied together (e.g., ), then some authors choose to abbreviate the Cartesian product as simply .

Cartesian product of functions

If is a function from to and is a function from to , then their Cartesian product is a function from to

This can be extended to tuples and infinite collections of functions. This is different from the standard Cartesian product of functions considered as sets.

Cylinder

. Then the cylinder.

is considered to be the universe.

Definitions outside set theory

Category theory

Although the Cartesian product is traditionally applied to sets, category theory provides a more general interpretation of the product the diagrams indexing products can be reduced to indexing sets matching the set-theoretic definition.

Graph theory

In graph theory, the Cartesian product of two graphs and is the graph denoted by , whose vertex set is the (ordinary) Cartesian product and such that two vertices and are adjacent in , if and only if and is adjacent with ′ in , or and is adjacent with ′ in . The Cartesian product of graphs is not a product in the sense of category theory. Instead, the categorical product is known as the tensor product of graphs.