In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B.^{[1]} In terms of set-builder notation, that is
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).^{[4]}
One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-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,^{[5]} whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
A rigorous definition of the Cartesian product requires a domain to be specified in the set-builder notation. In this case the domain would have to contain the Cartesian product itself. For defining the Cartesian product of the sets and , with the typical Kuratowski's definition of a pair (a,b) as , an appropriate domain is the set where denotes the power set. Then the Cartesian product of the sets and would be defined as^{[6]}
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.
Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), …, (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}.
Suits × Ranks 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.
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 x and y coordinates, respectively (see picture). The set of all such pairs (i.e., the Cartesian product ℝ×ℝ, with ℝ denoting the real numbers) is thus assigned to the set of all points in the plane.^{[citation needed]}
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, is . Under this definition, is an element of , and is a subset of that set, where 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.
Let A, B, C, and D be sets.
The Cartesian product A × B is not commutative,
because the ordered pairs are reversed unless at least one of the following conditions is satisfied:^{[7]}
For example:
Strictly speaking, the Cartesian product is not associative (unless one of the involved sets is empty).
If for example A = {1}, then (A × A) × A = {((1, 1), 1)} ≠ {(1, (1, 1))} = A × (A × A).
The Cartesian product satisfies the following property with respect to intersections (see middle picture).
In most cases, the above statement is not true if we replace intersection with union (see rightmost picture).
In fact, we have that:
For the set difference, we also have the following identity:
Here are some rules demonstrating distributivity with other operators (see leftmost picture):^{[7]}
where denotes the absolute complement of A.
Other properties related with subsets are:
The cardinality of a set is the number of elements of the set. For example, defining two sets: A = {a, b} and B = {5, 6}. Both set A and set B consist of two elements each. Their Cartesian product, written as A × B, results in a new set which has the following elements:
where each element of A is paired with each element of B, 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, |A × B| = 4
Similarly,
and so on.
The set A × B is infinite if either A or B is infinite, and the other set is not the empty set.^{[9]}
The Cartesian product can be generalized to the n-ary Cartesian product over n sets X_{1}, ..., X_{n} as the set
of n-tuples. If tuples are defined as nested ordered pairs, it can be identified with (X_{1} × ... × X_{n−1}) × X_{n}. If a tuple is defined as a function on {1, 2, ..., n} that takes its value at i to be the ith element of the tuple, then the Cartesian product X_{1} × ... × X_{n} is the set of functions
The Cartesian square of a set X is the Cartesian product X^{2} = X × X. An example is the 2-dimensional plane R^{2} = R × R where R is the set of real numbers:^{[1]} R^{2} is the set of all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).
The n-ary Cartesian power of a set X, denoted , can be defined as
An example of this is R^{3} = R × R × R, with R again the set of real numbers,^{[1]} and more generally R^{n}.
The n-ary Cartesian power of a set X is isomorphic to the space of functions from an n-element set to X. As a special case, the 0-ary Cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
It is possible to define the Cartesian product of an arbitrary (possibly infinite) indexed family of sets. If I is any index set, and is a family of sets indexed by I, then the Cartesian product of the sets in is defined to be
that is, the set of all functions defined on the index set I such that the value of the function at a particular index i is an element of X_{i}. Even if each of the X_{i} is nonempty, the Cartesian product may be empty if the axiom of choice, which is equivalent to the statement that every such product is nonempty, is not assumed. may also be denoted .^{[10]}
For each j in I, the function
defined by is called the jth projection map.
Cartesian power is a Cartesian product where all the factors X_{i} are the same set X. In this case,
is the set of all functions from I to X, and is frequently denoted X^{I}. This case is important in the study of cardinal exponentiation. An important special case is when the index set is , the natural numbers: this Cartesian product is the set of all infinite sequences with the ith term in its corresponding set X_{i}. For example, each element of
can be visualized as a vector with countably infinite real number components. This set is frequently denoted , or .
If several sets are being multiplied together (e.g., X_{1}, X_{2}, X_{3}, …), then some authors^{[11]} choose to abbreviate the Cartesian product as simply ×X_{i}.
If f is a function from X to A and g is a function from Y to B, then their Cartesian product f × g is a function from X × Y to A × B with
This can be extended to tuples and infinite collections of functions. This is different from the standard Cartesian product of functions considered as sets.
Let be a set and . Then the cylinder of with respect to is the Cartesian product of and .
Normally, is considered to be the universe of the context and is left away. For example, if is a subset of the natural numbers , then the cylinder of is .
Although the Cartesian product is traditionally applied to sets, category theory provides a more general interpretation of the product of mathematical structures. This is distinct from, although related to, the notion of a Cartesian square in category theory, which is a generalization of the fiber product.
Exponentiation is the right adjoint of the Cartesian product; thus any category with a Cartesian product (and a final object) is a Cartesian closed category.
In graph theory, the Cartesian product of two graphs G and H is the graph denoted by G × H, whose vertex set is the (ordinary) Cartesian product V(G) × V(H) and such that two vertices (u,v) and (u′,v′) are adjacent in G × H, if and only if u = u′ and v is adjacent with v′ in H, or v = v′ and u is adjacent with u′ in G. 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.