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Cartesian product
In mathematics, a Cartesian product is a mathematical operation which returns a set (or product set or simply product) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. Products can be specified using setbuilder notation, e.g.
 <math>A\times B = \{\,(a,b)\mid a\in A \ \mbox{ and } \ b\in B\,\}.</math>^{[1]}
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).
More generally, a Cartesian product of n sets, also known as a nfold Cartesian product, can be represented by an array of n dimensions, where each element is an ntuple. An ordered pair is a 2tuple or couple.
The Cartesian product is named after René Descartes,^{[2]} whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
Contents
Examples
A deck of cards
An illustrative example is the standard 52card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13element set. The card suits {♠, ♥, ♦, ♣} form a 4element set. The Cartesian product of these sets returns a 52element 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)}.
Both sets are distinct, even disjoint.
A twodimensional 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 component is called its x and y coordinate, respectively, cf. 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.
Most common implementation (set theory)
A formal definition of the Cartesian product from settheoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, the Kuratowski definition, is <math>(x, y) = \{\{x\},\{x, y\}\}</math>. Note that, under this definition, <math>X\times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y))</math>, where <math>\mathcal{P}</math> represents the power set. 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 twoset Cartesian product is necessarily prior to most other definitions.
Noncommutativity and nonassociativity
Let A, B, C, and D be sets.
The Cartesian product A × B is not commutative,
 <math>A \times B \neq B \times A,</math>
because the ordered pairs are reversed except if at least one of the following conditions is satisfied:^{[3]}
 A is equal to B, or
 A or B is the empty set.
For example:
 A = {1,2}; B = {3,4}
 A × B = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
 B × A = {3,4} × {1,2} = {(3,1), (3,2), (4,1), (4,2)}
 A = B = {1,2}
 A × B = B × A = {1,2} × {1,2} = {(1,1), (1,2), (2,1), (2,2)}
 A = {1,2}; B = ∅
 A × B = {1,2} × ∅ = ∅
 B × A = ∅ × {1,2} = ∅
Strictly speaking, the Cartesian product is not associative (unless one of the involved sets is empty).
 <math>(A\times B)\times C \neq A \times (B \times C)</math>
If for example A = {1}, then (A × A) × A = { ((1,1),1) } ≠ { (1,(1,1)) } = A × (A × A).
Intersections, unions, and subsets
The Cartesian product behaves nicely with respect to intersections, cf. left picture.
 <math>(A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D)</math>^{[4]}
In most cases the above statement is not true if we replace intersection with union, cf. middle picture.
 <math>(A \cup B) \times (C \cup D) \neq (A \times C) \cup (B \times D)</math>
In fact, we have that:
 <math>(A \times C) \cup (B \times D) = [(A \setminus B) \times C] \cup [(A \cap B) \times (C \cup D)] \cup [(B \setminus A) \times D]</math>
For the set difference we also have the following identity:
 <math>(A \times C) \setminus (B \times D) = [A \times (C \setminus D)] \cup [(A \setminus B) \times C] </math>
Here are some rules demonstrating distributivity with other operators (cf. right picture):^{[3]}
 <math>A \times (B \cap C) = (A \times B) \cap (A \times C),</math>
 <math>A \times (B \cup C) = (A \times B) \cup (A \times C),</math>
 <math>A \times (B \setminus C) = (A \times B) \setminus (A \times C),</math>
 <math>(A \times B)^c = (A^c \times B^c) \cup (A^c \times B) \cup (A \times B^c).</math>^{[4]}
Other properties related with subsets are:
 <math>\text{if } A \subseteq B \text{ then } A \times C \subseteq B \times C,</math>
 <math>\text{if both } A,B \neq \emptyset \text{ then } A \times B \subseteq C \times D \iff A \subseteq C \and B \subseteq D.</math>^{[5]}
Cardinality
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:
 A × B = {(a,5), (a,6), (b,5), (b,6)}.
Each element of A is paired with each element of B. 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,
 A × B = A · B.
Similarly
 A × B × C = A · B · C
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.^{[6]}
nary product
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 2dimensional plane R^{2} = R × R where R is the set of real numbers: R^{2} is the set of 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:
 <math> X^n = \underbrace{ X \times X \times \cdots \times X }_{n}= \{ (x_1,\ldots,x_n) \  \ x_i \in X \text{ for all } i=1,\ldots,n \}.</math>
An example of this is R^{3} = R × R × R, with R again the set of real numbers, and more generally R^{n}.
The nary cartesian power of a set X is isomorphic to the space of functions from an nelement set to X. As a special case, the 0ary cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
Finite nary product
The Cartesian product can be generalized to the nary Cartesian product over n sets X_{1}, ..., X_{n}:
 <math>X_1\times\cdots\times X_n = \{(x_1, \ldots, x_n) : x_i \in X_i \}.</math>
It is a set of ntuples. If tuples are defined as nested ordered pairs, it can be identified to (X_{1} × ... × X_{n−1}) × X_{n}.
Infinite products
It is possible to define the Cartesian product of an arbitrary (possibly infinite) indexed family of sets. If I is any index set, and <math>\left\{X_i\,\,i\in I\right\}</math> is a collection of sets indexed by I, then the Cartesian product of the sets in X is defined to be
 <math>\prod_{i \in I} X_i = \left\{ f : I \to \bigcup_{i \in I} X_i\ \Big\ (\forall i)(f(i) \in X_i)\right\},</math>
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}. 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.
For each j in I, the function
 <math> \pi_{j} : \prod_{i \in I} X_i \to X_{j},</math>
defined by <math>\pi_{j}(f) = f(j)</math> is called the jth projection map.
An important case is when the index set is <math>\mathbb{N}</math>, 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
 <math>\prod_{n = 1}^\infty \mathbb R = \mathbb R \times \mathbb R \times \cdots</math>
can be visualized as a vector with countably infinite real number components. This set is frequently denoted <math>\mathbb{R}^\omega</math>, or <math>\mathbb{R}^{\mathbb{N}}</math>.
The special case Cartesian exponentiation occurs when all the factors X_{i} involved in the product are the same set X. In this case,
 <math>\prod_{i \in I} X_i = \prod_{i \in I} X</math>
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.
The definition of finite Cartesian products can be seen as a special case of the definition for infinite products. In this interpretation, an ntuple can be viewed as a function on {1, 2, ..., n} that takes its value at i to be the ith element of the tuple (in some settings, this is taken as the very definition of an ntuple).
Other forms
Abbreviated form
If several sets are being multiplied together, e.g. X_{1}, X_{2}, X_{3}, …, then some authors^{[7]} choose to abbreviate the Cartesian product as simply ×X_{i}.
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
 <math>(f\times g)(a, b) = (f(a), g(b)).</math>
This can be extended to tuples and infinite collections of functions. Note that this is different from the standard cartesian product of functions considered as sets.
Definitions outside of Set theory
Category theory
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.
Graph theory
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.
See also
 Exponential object
 Binary relation
 Coproduct
 Empty product
 Product (category theory)
 Concatenation of sets is deceptively similar but different concept
 Product topology
 Finitary relation
 Ultraproduct
 Product type
 Euclidean space
 Orders on R^{n}
 Join (SQL), § Cross join
References
 ^ Warner, S: Modern Algebra, page 6. Dover Press, 1990.
 ^ cartesian. (2009). In MerriamWebster Online Dictionary. Retrieved December 1, 2009, from http://www.merriamwebster.com/dictionary/cartesian
 ^ ^{a} ^{b} Singh, S. (2009, August 27). Cartesian product. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/
 ^ ^{a} ^{b} CartesianProduct at PlanetMath.org.
 ^ Cartesian Product of Subsets. (2011, February 15). ProofWiki. Retrieved 05:06, August 1, 2011 from https://proofwiki.org/w/index.php?title=Cartesian_Product_of_Subsets&oldid=45868
 ^ Peter S. (1998). A Crash Course in the Mathematics Of Infinite Sets. St. John's Review, 44(2), 35–59. Retrieved August 1, 2011, from http://www.mathpath.org/concepts/infinity.htm
 ^ Osborne, M., and Rubinstein, A., 1994. A Course in Game Theory. MIT Press.
External links
 Cartesian Product at ProvenMath
 Hazewinkel, Michiel, ed. (2001), "Direct product", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 How to find the Cartesian Product, Education Portal Academy
