Cartesian product

Last updated

Cartesian product
{\displaystyle \scriptstyle A\times B}
of the sets
{\displaystyle \scriptstyle A=\{x,y,z\}}
{\displaystyle \scriptstyle B=\{1,2,3\}} Cartesian Product qtl1.svg
Cartesian product of the sets and

In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A×B, [1] is the set of all ordered pairs (a, b) where a is in A and b is in B. [2] In terms of set-builder notation, that is


[3] [4]

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). [5]

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, [6] whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.


A deck of cards

Standard 52-card deck Piatnikcards.jpg
Standard 52-card deck

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.

A two-dimensional coordinate system

Cartesian coordinates of example points Cartesian-coordinate-system.svg
Cartesian coordinates of example points

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 ]

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, 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.

Non-commutativity and non-associativity

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:

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).

If for example A = {1}, then (A × A) × A = {((1, 1), 1)} ≠{(1, (1, 1))} = A × (A × A).

Intersections, unions, and subsets

CartDistr svg.svg
Example sets

A={y  :1y4}, B={x : 2x5},
and C = {x : 4x7}, demonstrating
A × (BC) = (A×B) ∩ (A×C),
A × (BC) = (A×B) ∪ (A×C), and

A × (B\C) = (A×B) \ (A×C)
CartInts svg.svg
Example sets

A={x : 2x5}, B={x : 3x7},
C={y :1y3}, D={y : 2y4}, demonstrating

(AB) × (CD) = (A×C) ∩ (B×D).
CartUnion svg.svg
(AB) × (CD) ≠ (A×C) ∪ (B×D) can be seen from the same example.

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:

A × B = {(a,5), (a,6), (b,5), (b,6)}.

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,

|A × B| = |A| · |B|. [5]

In this case, |A × B| = 4


|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. [10]

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

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

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: [2] 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, denoted , [1] can be defined as

An example of this is R3 = R × R × R, with R again the set of real numbers, [2] 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 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 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

defined by 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,

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 , 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

can be visualized as a vector with countably infinite real number components. This set is frequently denoted , or .

Other forms

Abbreviated form

If several sets are being multiplied together (e.g., X1, X2, X3, …), then some authors [11] 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, then their Cartesian product f × g is a function from A × X to B × Y 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 .

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.

See also

Related Research Articles

In mathematics, a binary relation over sets X and Y is a subset of the Cartesian product ; that is, it is a set of ordered pairs (x, y) consisting of elements x in X and y in Y. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case n = 2 of an n-ary relation over sets X1, ..., Xn, which is a subset of the Cartesian product

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.

In mathematics, a directed set is a nonempty set together with a reflexive and transitive binary relation , with the additional property that every pair of elements has an upper bound. In other words, for any and in there must exist in with and A directed set's preorder is called a direction.

In mathematics, the inverse limit is a construction that generalizes several other constructions that recur throughout various fields of mathematics, including products, pullbacks, intersections, and an endless amount of other related constructions, only some of which have been investigated − profinite groups, p-adic numbers, solenoids, and ring completions being some prominent examples. Inverse limits are taken of inverse systems in some given category, such as the category Set of sets for example. An inverse system is a collection of objects in the category indexed by some preordered set called the indexing set, together with a collection of morphisms called connecting morphisms that satisfy the following compatibility condition of inverse systems: A cone into this inverse system is pair consisting of an object called its vertex, and an -indexed family of morphisms each of the form and satisfying the compatibility condition: An inverse limit of this inverse system, if it exists, is a cone that also has an additional property known as the universal property of inverse limits. This universal property establishes a one-to-one correspondence between, on one hand, cones into the given inverse system and, on the other hand, a unique morphism into the limit object that is "compatible" with the limit's family of morphisms in that it satisfies the condition: .

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.

Partially ordered set Set ordered by a transitive, antisymmetric, and reflexive binary relation

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word partial in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.

In mathematics, a product is the result of multiplication, or an expression that identifies factors to be multiplied. For example, 30 is the product of 6 and 5, and is the product of and .

Set (mathematics) Collection of mathematical objects

In mathematics a set is a collection of distinct elements. The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets. Two sets are equal if and only if they have precisely the same elements.

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

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.

In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory.

In database theory, relational algebra is a theory that uses algebraic structures with a well-founded semantics for modeling data, and defining queries on it. The theory has been introduced by Edgar F. Codd.

Function (mathematics) Mapping that associates a single output value to each input

In mathematics, a function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

Symmetric difference Subset of the elements that belong to exactly one among two sets

In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets and is .

Restriction (mathematics)

In mathematics, the restriction of a function is a new function, denoted or , obtained by choosing a smaller domain A for the original function .

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 π-system on a set is a collection of certain subsets of such that

In mathematical analysis its applications, 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.

Filters in topology Use of filters to describe and characterize all basic topological notions and results.

In topology, a subfield of mathematics, filters are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.


  1. 1 2 "Comprehensive List of Set Theory Symbols". Math Vault. April 11, 2020. Retrieved September 5, 2020.
  2. 1 2 3 Weisstein, Eric W. "Cartesian Product". Retrieved September 5, 2020.
  3. Warner, S. (1990). Modern Algebra. Dover Publications. p. 6.
  4. Nykamp, Duane. "Cartesian product definition". Math Insight. Retrieved September 5, 2020.
  5. 1 2 3 "Cartesian Product". Retrieved September 5, 2020.
  6. "Cartesian". 2009. Retrieved December 1, 2009.
  7. 1 2 Singh, S. (August 27, 2009). Cartesian product. Retrieved from the Connexions Web site:
  8. 1 2 "CartesianProduct". PlanetMath .
  9. Cartesian Product of Subsets. (February 15, 2011). ProofWiki. Retrieved 05:06, August 1, 2011 from
  10. 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
  11. Osborne, M., and Rubinstein, A., 1994. A Course in Game Theory. MIT Press.