Cartesian product

Cartesian product of the sets {x,y,z} and {1,2,3}

In mathematics, specifically set theory, the Cartesian product of two sets Template:Mvar and Template:Mvar, denoted Template:Math, is the set of all ordered pairs Template:Math where Template:Mvar is in Template:Mvar and Template:Mvar is in Template:Mvar. In terms of set-builder notation, that is $$A\times B=\{(a,b)\mid a\in A\text{ and }b\in B\}.$$

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

One can similarly define the Cartesian product of Template:Mvar sets, also known as an Template:Mvar-fold Cartesian product, which can be represented by an Template:Mvar-dimensional array, where each element is an Template:Mvar-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. In this case the domain would have to contain the Cartesian product itself. For defining the Cartesian product of the sets $A$ and $B$, with the typical Kuratowski's definition of a pair $(a,b)$ as $\{\{a\},\{a,b\}\}$, an appropriate domain is the set $𝒫(𝒫(A\cup B))$ where $𝒫$ denotes the power set. Then the Cartesian product of the sets $A$ and $B$ would be defined as $$A\times B=\{x\in 𝒫(𝒫(A\cup B))\mid \mathrm{\exists }a\in A\mathrm{\exists }b\in B:x=(a,b)\}.$$

Examples

A deck of cards

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 Template:Nowrap} 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.

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

Template:Nowrap 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

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 Template:Mvar and Template:Mvar coordinates, respectively (see picture). The set of all such pairs (i.e., the Cartesian product $ℝ\times ℝ$, 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 set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, Kuratowski's definition, is $(x,y)=\{\{x\},\{x,y\}\}$. Under this definition, $(x,y)$ is an element of $𝒫(𝒫(X\cup Y))$, and $X\times Y$ is a subset of that set, where $𝒫$ 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.

Non-commutativity and non-associativity

Let Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar be sets.

The Cartesian product Template:Math is not commutative, $$A\times B\ne B\times A,$$ because the ordered pairs are reversed unless at least one of the following conditions is satisfied:

• Template:Mvar is equal to Template:Mvar, or
• Template:Mvar or Template:Mvar is the empty set.

For example:

Template:Math; Template:Math

Template:Math

Template:Math

Template:Math

Template:Math

Template:Math

Template:Math

Template:Math

Strictly speaking, the Cartesian product is not associative (unless one of the involved sets is empty). $$(A\times B)\times C\ne A\times (B\times C)$$ If for example Template:Math, then Template:Math Template:Math.

Intersections, unions, and subsets

The Cartesian product satisfies the following property with respect to intersections (see middle 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 rightmost picture). $$(A\cup B)\times (C\cup D)\ne (A\times C)\cup (B\times D)$$

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 leftmost picture): $$\begin{array}{cc}A\times (B\cap C)& =(A\times B)\cap (A\times C),\\ A\times (B\cup C)& =(A\times B)\cup (A\times C),\\ A\times (B\setminus C)& =(A\times B)\setminus (A\times C),\end{array}$$ $$(A\times B{)}^{\complement }=(A^{\complement }\times B^{\complement })\cup (A^{\complement }\times B)\cup (A\times B^{\complement }),$$ where $A^{\complement }$ denotes the absolute complement of Template:Mvar.

Other properties related with subsets are:

$$\text{if }A\subseteq B\text{, then }A\times C\subseteq B\times C;$$

$$\text{if both }A,B\ne \mathrm{\varnothing }\text{, then }A\times B\subseteq C\times D⟺A\subseteq C\text{ and }B\subseteq D.$$

Cardinality

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

Template:Math.

where each element of Template:Mvar is paired with each element of Template:Mvar, 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,

Template:Math.

In this case, Template:Math

Similarly,

Template:Math

and so on.

The set Template:Math is infinite if either Template:Mvar or Template:Mvar 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 Template:Mvar-ary Cartesian product over Template:Mvar sets Template:Math as the set $$X_1\times \cdots \times X_n=\{(x_1,\dots ,x_n)\mid x_i\in X_i\text{for every}i\in \{1,\dots ,n\}\}$$

of [[tuple|Template:Mvar-tuple]]s. If tuples are defined as nested ordered pairs, it can be identified with Template:Math. If a tuple is defined as a function on Template:Math} that takes its value at Template:Mvar to be the Template:Mvar-th element of the tuple, then the Cartesian product Template:Math is the set of functions $$\{x:\{1,\dots ,n\}\to X_1\cup \cdots \cup X_n|x(i)\in X_i\text{for every}i\in \{1,\dots ,n\}\}.$$

n-ary Cartesian power

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

The Template:Mvar-ary Cartesian power of a set Template:Mvar, denoted $X^n$, can be defined as $$X^n=\underset{n}{\underbrace{X\times X\times \cdots \times X}}=\{(x_1,\dots ,x_n)|x_i\in X\text{for every}i\in \{1,\dots ,n\}\}.$$

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

The Template:Mvar-ary Cartesian power of a set Template:Mvar is isomorphic to the space of functions from an Template:Mvar-element set to Template:Mvar. As a special case, the 0-ary Cartesian power of Template:Mvar may be taken to be a singleton set, corresponding to the empty function with codomain Template:Mvar.

Intersections, unions, complements and subsets

Let Cartesian products be given $A=A_1\times \dots \times A_n$ and $B=B_1\times \dots \times B_n$. Then

1. $A\subseteq B$, if and only if $A_i\subseteq B_i$ for all $i=1,2,\dots ,n$;
2. $A\cap B=(A_1\cap B_1)\times \dots \times (A_n\cap B_n)$, at the same time, if there exists at least one $i$ such that $A_i\cap B_i=\varnothing$, then $A\cap B=\varnothing$;
3. $A\cup B\subseteq (A_1\cup B_1)\times \dots \times (A_n\cup B_n)$, moreover, equality is possible only in the following cases:
   1. $A\subseteq B$ or $B\subseteq A$;
   2. for all $i=1,2,\dots ,n{A}_i=B_i$ except for one from $i$.
4. The complement of a Cartesian product $A=A_1\times \dots \times A_n$ can be calculated, if a universe is defined $U=X_1\times \dots \times X_n$. 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.:

$A=A_1\times A_2\times \dots \times A_n=[A_1{A}_2\dots A_n]$.

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:

$A\cup B=(A_1\times A_2\times \dots \times A_n)\cup (B_1\times B_2\times \dots \times B_n)=\left[\begin{array}{cccc}{A}_1& A_2& \dots & A_n\\ B_1& B_2& \dots & B_n\end{array}\right]$.

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

Then the complement of the Cartesian product $A$ will look like the following C-system expressed as a matrix of the dimension $n\times n$:

$A^{\complement }=\left[\begin{array}{ccccc}{A}_1^{\complement }& X_2& \dots & X_{n-1}& X_n\\ X_1& A_2^{\complement }& \dots & X_{n-1}& X_n\\ \dots & \dots & \dots & \dots & \dots \\ X_1& X_2& \dots & A_{n-1}^{\complement }& X_n\\ X_1& X_2& \dots & X_{n-1}& A_n^{\complement }\end{array}\right]$.

The diagonal components of this matrix $A_i^{\complement }$ are equal correspondingly to $X_i\setminus A_i$.

In NTA, a diagonal C-system $A^{\complement }$, that represents the complement of a C-n-tuple $A$, can be written concisely as a tuple of diagonal components bounded by inverted square brackets:

$A^{\complement }=]A_1^{\complement }{A}_2^{\complement }\dots A_n^{\complement }[$.

This structure is called a D-n-tuple. Then the complement of the C-system $R$ is a structure $R^{\complement }$, represented by a matrix of the same dimension and bounded by inverted square brackets, in which all components are equal to the complements of the components of the initial matrix $R$. 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:

$R_1=\left[\begin{array}{cccc}{A}_1& A_2& \dots & A_n\\ B_1& B_2& \dots & B_n\end{array}\right]$,

then its complement will be the D-system

$R_1^{\complement }=\left]\begin{array}{cccc}{A}_1^{\complement }& A_2^{\complement }& \dots & A_n^{\complement }\\ B_1^{\complement }& B_2^{\complement }& \dots & B_n^{\complement }\end{array}\right[$.

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.

1. The intersection of C-systems. Assume the homotypic C-systems are given $P$ and $Q$. Their intersection will yield a C-system containing all non-empty intersections of each C-n-tuple from $P$ with each C-n-tuple from $Q$.
2. Checking the inclusion of a C-n-tuple into a D-n-tuple. For the C-n-tuple $P=[P_1{P}_2\cdots P_N]$ and the D-n-tuple $Q=]Q_1{Q}_2\cdots Q_N[$ holds $P\subseteq Q$, if and only if, at least for one $i$ holds $P_i\subseteq Q_i$.
3. Checking the inclusion of a C-n-tuple into a D-system. For the C-n-tuple $P$ and the D-system $Q$ is true $P\subseteq Q$, if and only if, for every D-n-tuple $Q_i$ from $Q$ holds $P\subseteq Q_i$.

Infinite Cartesian products

It is possible to define the Cartesian product of an arbitrary (possibly infinite) indexed family of sets. If Template:Mvar is any index set, and $\{X_i{\}}_{i\in I}$ is a family of sets indexed by Template:Mvar, then the Cartesian product of the sets in $\{X_i{\}}_{i\in I}$ is defined to be $$\prod _{i\in I}{X}_i=\{f:I\to \bigcup _{i\in I}{X}_i|\mathrm{\forall }i\in I.f(i)\in X_i\},$$ that is, the set of all functions defined on the index set Template:Mvar such that the value of the function at a particular index Template:Mvar 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. $\prod _{i\in I}{X}_i$ may also be denoted $𝖷$${}_{i\in I}{X}_i$.

For each Template:Mvar in Template:Mvar, the function $${\pi }_j:\prod _{i\in I}{X}_i\to X_j,$$ defined by ${\pi }_j(f)=f(j)$ is called the Template:Mvar-th projection map.

Cartesian power is a Cartesian product where all the factors X_i are the same set Template:Mvar. In this case, $$\prod _{i\in I}{X}_i=\prod _{i\in I}X$$ is the set of all functions from Template:Mvar to Template:Mvar, 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 $\mathbb{N}$, the natural numbers: this Cartesian product is the set of all infinite sequences with the Template:Mvar-th term in its corresponding set X_i. For example, each element of $$\prod _{n=1}^{\mathrm{\infty }}ℝ=ℝ\times ℝ\times \cdots$$ can be visualized as a vector with countably infinite real number components. This set is frequently denoted ${ℝ}^{\omega }$, or ${ℝ}^{\mathbb{N}}$.

Other forms

Abbreviated form

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

Cartesian product of functions

If Template:Mvar is a function from Template:Mvar to Template:Mvar and Template:Mvar is a function from Template:Mvar to Template:Mvar, then their Cartesian product Template:Math is a function from Template:Math to Template:Math with $$(f\times g)(x,y)=(f(x),g(y)).$$

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 Template:Mvar and Template:Mvar is the graph denoted by Template:Math, whose vertex set is the (ordinary) Cartesian product Template:Math and such that two vertices Template:Math and Template:Math are adjacent in Template:Math, if and only if Template:Math and Template:Mvar is adjacent with Template:Mvar′ in Template:Mvar, or Template:Math and Template:Mvar is adjacent with Template:Mvar′ in Template:Mvar. 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

• Axiom of power set (to prove the existence of the Cartesian product)
• Direct product
• Empty product
• Finitary relation
• Join (SQL) § Cross join
• Orders on the Cartesian product of totally ordered sets
• Outer product
• Product (category theory)
• Product topology
• Product type

External links

• Cartesian Product at ProvenMath
• Template:SpringerEOM
• How to find the Cartesian Product, Education Portal Academy