Cartesian product

Last updated Cartesian product A×B{\displaystyle \scriptstyle A\times B} of the sets A={x,y,z}{\displaystyle \scriptstyle A=\{x,y,z\}} and B={1,2,3}{\displaystyle \scriptstyle B=\{1,2,3\}}

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that 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 aA and bB. Products can be specified using set-builder notation, e.g. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects. Mathematics includes the study of such topics as quantity, structure, space, and change. It has no generally accepted definition. In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2, 4, 6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

Contents

$A\times B=\{\,(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\,\}.$ 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 an n-fold Cartesian product, can be represented by an array of n dimensions, where each element is an n-tuple. An ordered pair is a 2-tuple or couple.

In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence of n elements, where n is a non-negative integer. There is only one 0-tuple, an empty sequence, or empty tuple, as it is referred to. An n-tuple is defined inductively using the construction of an ordered pair.

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. René Descartes was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–1649) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. One of the most notable intellectual figures of the Dutch Golden Age, Descartes is also widely regarded as one of the founders of modern philosophy. In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.

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. The standard 52-card deck of French playing cards is the most common deck of playing cards used today. It includes thirteen ranks in each of the four French suits: clubs, diamonds, hearts and spades, with reversible "court" or face cards. Each suit includes an ace, a king, queen and jack, each depicted with a symbol of its suit; and ranks two through ten, with each card depicting that many symbols (pips) of its suit. Anywhere from one to six jokers, often distinguishable with one being more colorful than the other, are added to commercial decks, as some card games require these extra cards. Modern playing cards carry index labels on opposite corners or in all four corners to facilitate identifying the cards when they overlap and so that they appear identical for players on opposite sides. The most popular standard pattern of the French deck is sometimes referred to as "English" or "Anglo-American" pattern.

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 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 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. In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as 2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

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, the Kuratowski definition, is $(x,y)=\{\{x\},\{x,y\}\}$ . Under this definition, $(x,y)$ is an element of ${\mathcal {P}}({\mathcal {P}}(X\cup Y))$ , and $X\times Y$ is a subset of that set, where ${\mathcal {P}}$ 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.

In mathematics, an ordered pair is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair is different from the ordered pair unless a = b. In mathematics, the power set of any set S is the set of all subsets of S, including the empty set and S itself, variously denoted as P(S), 𝒫(S), ℘(S), P(S), ℙ(S), or, identifying the powerset of S with the set of all functions from S to a given set of two elements, 2S. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by Zermelo (1908) as a special case of his axiom of elementary sets.

Non-commutativity and non-associativity

Let A, B, C, and D be sets.

The Cartesian product A × B is not commutative,

$A\times B\neq B\times A,$ because the ordered pairs are reversed unless at least one of the following conditions is satisfied: 

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).

$(A\times B)\times C\neq A\times (B\times C)$ If for example A = {1}, then (A × A) × A = { ((1,1),1) } ≠ { (1,(1,1)) } = A × (A × A).

Intersections, unions, and subsets

Example sets

A={y:1≤y≤4},
B={x∈ℝ:2≤x≤5}, and C={x∈ℝ:4≤x≤7}, demonstrating
A×(BC) = (A×B)∩(A×C),
A×(BC) = (A×B)∪(A×C), and

A×(B\C) = (A×B)\(A×C)
Example sets

A={x∈ℝ:2≤x≤5}, B={x∈ℝ:3≤x≤7},
C={y∈ℝ:1≤y≤3}, D={y∈ℝ:2≤y≤4}, demonstrating

(AB)×(CD) = (A×C)∩(B×D).

The Cartesian product behaves nicely with respect to intersections (see leftmost picture).

$(A\cap B)\times (C\cap D)=(A\times C)\cap (B\times D)$ In most cases the above statement is not true if we replace intersection with union (see middle picture).

$(A\cup B)\times (C\cup D)\neq (A\times C)\cup (B\times D)$ In fact, we have that:

$(A\times C)\cup (B\times D)=[(A\setminus B)\times C]\cup [(A\cap B)\times (C\cup D)]\cup [(B\setminus A)\times D]$ For the set difference we also have the following identity:

$(A\times C)\setminus (B\times D)=[A\times (C\setminus D)]\cup [(A\setminus B)\times C]$ Here are some rules demonstrating distributivity with other operators (see rightmost picture): 

$A\times (B\cap C)=(A\times B)\cap (A\times C),$ $A\times (B\cup C)=(A\times B)\cup (A\times C),$ $A\times (B\setminus C)=(A\times B)\setminus (A\times C),$ $(A\times B)^{\complement }=(A^{\complement }\times B^{\complement })\cup (A^{\complement }\times B)\cup (A\times B^{\complement }),$ where $A^{\complement }$ denotes the absolute complement of A.

Other properties related with subsets are:

${\text{if }}A\subseteq B{\text{ then }}A\times C\subseteq B\times C,$ ${\text{if both }}A,B\neq \emptyset {\text{ then }}A\times B\subseteq C\times D\iff A\subseteq C{\text{ and }}B\subseteq D.$ 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|.

In this case, |A × B| = 4

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. 

Cartesian products of several sets

n-ary Cartesian product

The Cartesian product can be generalized to the n-ary Cartesian product over n sets X1, ..., Xn as the set

$X_{1}\times \cdots \times X_{n}=\{(x_{1},\ldots ,x_{n})\mid x_{i}\in X_{i}\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.$ of n-tuples. If tuples are defined as nested ordered pairs, it can be identified with (X1 × ... × Xn−1) × Xn. 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 X1×...×Xn is the set of functions

$\{x:\{1,\ldots ,n\}\to X_{1}\cup \ldots \cup X_{n}\ |\ x(i)\in X_{i}\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.$ n-ary Cartesian power

The Cartesian square of a set X is the Cartesian product X2 = X × X. An example is the 2-dimensional plane R2 = R × R where R is the set of real numbers: R2 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 can be defined as

$X^{n}=\underbrace {X\times X\times \cdots \times X} _{n}=\{(x_{1},\ldots ,x_{n})\ |\ x_{i}\in X\ {\text{for every}}\ i\in \{1,\ldots ,n\}\}.$ An example of this is R3 = R × R × R, with R again the set of real numbers, and more generally Rn.

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.

Infinite Cartesian 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 $\{X_{i}\}_{i\in I}$ is a family of sets indexed by I, then the Cartesian product of the sets in X is defined to be

$\prod _{i\in I}X_{i}=\left\{\left.f:I\to \bigcup _{i\in I}X_{i}\ \right|\ (\forall i)(f(i)\in X_{i})\right\},$ 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 Xi. Even if each of the Xi 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

$\pi _{j}:\prod _{i\in I}X_{i}\to X_{j},$ defined by $\pi _{j}(f)=f(j)$ is called the jth projection map .

Cartesian power is a Cartesian product where all the factors Xi are the same set X. In this case,

$\prod _{i\in I}X_{i}=\prod _{i\in I}X$ is the set of all functions from I to X, and is frequently denoted XI. This case is important in the study of cardinal exponentiation. An important special case is when the index set is $\mathbb {N}$ , the natural numbers: this Cartesian product is the set of all infinite sequences with the ith term in its corresponding set Xi. For example, each element of

$\prod _{n=1}^{\infty }\mathbb {R} =\mathbb {R} \times \mathbb {R} \times \cdots$ can be visualized as a vector with countably infinite real number components. This set is frequently denoted $\mathbb {R} ^{\omega }$ , or $\mathbb {R} ^{\mathbb {N} }$ .

Other forms

Abbreviated form

If several sets are being multiplied together, e.g. X1, X2, X3, …, then some authors  choose to abbreviate the Cartesian product as simply ×Xi.

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

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

Let $A$ be a set and $B\subseteq A$ . Then the cylinder of $B$ with respect to $A$ is the Cartesian product $B\times A$ of $B$ and $A$ .

Normally, $A$ is considered to be the universe of the context and is left away. For example, if $B$ is a subset of the natural numbers $\mathbb {N}$ , then the cylinder of $B$ is $B\times \mathbb {N}$ .

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 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, orv = 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.

Related Research Articles

In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance, 6 is the product of 2 and 3, and is the product of and . In set theory, the complement of a set A refers to elements not in A.

In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.

In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. It was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory.

Relational algebra, first created by Edgar F. Codd while at IBM, is a family of algebras with a well-founded semantics used for modelling the data stored in relational databases, and defining queries on it. In mathematics, a function is a relation between sets, that associates to every element of a first set exactly one element of the second set. Typical examples are functions from integers to integers or from the real numbers to real numbers.

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis. In mathematics, the symmetric difference, also known as the disjunctive union, of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by

In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.

In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient is the set In mathematics, the restriction of a functionf is a new function obtained by choosing a smaller domain A for the original function . The notation is also used.

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set satisfying a set of axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems or d-system. These set families have applications in measure theory and probability.

This article examines the implementation of mathematical concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC and in NFU, the version of Quine's New Foundations shown to be consistent by R. B. Jensen in 1969.

In mathematics, a submodular set function is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article.

In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf of commutative ring spectra on X such that (1) the pair is a scheme and (2) is a quasi-coherent -module. The notion gives a homotopy-theoretic generalization of a scheme.

1. Warner, S. (1990). Modern Algebra. Dover Publications. p. 6.
2. "Cartesian". Merriam-Webster.com. 2009. Retrieved December 1, 2009.
3. Singh, S. (August 27, 2009). Cartesian product. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/
4. Cartesian Product of Subsets. (February 15, 2011). ProofWiki. Retrieved 05:06, August 1, 2011 from https://proofwiki.org/w/index.php?title=Cartesian_Product_of_Subsets&oldid=45868
5. 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
6. Osborne, M., and Rubinstein, A., 1994. A Course in Game Theory. MIT Press.