Cartesian product

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 an element of A and b is an element of B. In terms of set-builder notation, that is \[A\times B = \{(a,b)\mid a \in A \ \mbox{ and } \ b \in B\}.\]

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).

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, 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 \(A\) and \(B\), with the typical Kuratowski's definition of a pair \((a,b)\) as \(\{\{a\},\{a,b\}\}\), an appropriate domain is the set \(\mathcal{P}(\mathcal{P}(A\cup B))\) where \(\mathcal{P}\) denotes the power set. Then the Cartesian product of the sets \(A\) and \(B\) would be defined as \[A\times B=\{x\in\mathcal{P}(\mathcal{P}(A\cup B))\mid\exists a\in A\ \exists b\in B:x=(a,b)\}.\]

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 \(\R\times\R\), with \(\R\) denoting the real numbers) is thus assigned to the set of all points in the plane.

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 \(\mathcal{P}(\mathcal{P}(X \cup Y))\), 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.

Let A, B, and C 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:

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

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). \[(A \cup B) \times (C \cup D) \neq (A \times C) \cup (B \times D)\]

In fact, we have that: \[(A \times C) \cup (B \times D) = [(A \smallsetminus B) \times C] \cup [(A \cap B) \times (C \cup D)] \cup [(B \smallsetminus A) \times D]\]

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

Here are some rules demonstrating distributivity with other operators (see leftmost picture): \[\begin{aligned} A \times (B \cap C) &= (A \times B) \cap (A \times C), \\ A \times (B \cup C) &= (A \times B) \cup (A \times C), \\ A \times (B \smallsetminus C) &= (A \times B) \smallsetminus (A \times C), \end{aligned}\] \[(A \times B)^\complement = \left(A^\complement \times B^\complement\right) \cup \left(A^\complement \times B\right) \cup \left(A \times B^\complement\right)\!,\] where \(A^\complement\) denotes the absolute complement of A.

Other properties related with subsets are:

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

\[\text{if both } A,B \neq \emptyset \text{, then } A \times B \subseteq C \times D \!\iff\! A \subseteq C \text{ and } B \subseteq D.\]

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|.

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.

The Cartesian product can be generalized to the n-ary Cartesian product over n sets X₁, ..., X_n 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 × ... × 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 i-th element of the tuple, then the Cartesian product X₁ × ... × X_n is the set of functions \[\{ x:\{1,\ldots,n\}\to X_1\cup\cdots\cup X_n \mid x(i)\in X_i \text{ for every } i \in \{1, \ldots, n\} \}.\]

The Cartesian square of a set X is the Cartesian product X = X × X. An example is the 2-dimensional plane R = R × R where R is the set of real numbers: R is the set of all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).

The Cartesian nth power of a set X, denoted \(X^n\), can be defined as \[X^n = \underbrace{ X \times X \times \cdots \times X}_{n}= \{ (x_1,\ldots,x_n) \mid x_i \in X \text{ for every } i \in \{1, \ldots, n\} \}.\]

An example of this is R = R × R × R, with R again the set of real numbers, and more generally R.

The Cartesian nth power of a set X may be identified with the set of the functions mapping to X the n-tuples of elements of X. As a special case, the Cartesian 0th power of X is the singleton set, that has the empty function with codomain X as its unique element.

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

1. \(A \sube B\), if and only if \(A_i \sube B_i\) for all \(i = 1, 2,\ldots, n\);
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. \(A \cup B \sube (A_1 \cup B_1) \times \dots \times (A_n \cup B_n)\), moreover, equality is possible only in the following cases:
   1. \(A \sube B\) or \(B \sube A\);
   2. for all \(i = 1, 2,\ldots, n \quad A_i = B_i \quad\) 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.:

\(A = A_1 \times A_2 \times \dots \times A_n =[A_1\quad A_2\quad \dots \quad A_n]\).

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 \(n \times n\):

\(A^\complement =\left [\begin{array} {ccccc} A_1^\complement & X_2 & \cdots & X_{n-1} & X_n \\ X_1 & A_2^\complement & \cdots & X_{n-1} & X_n \\ \vdots & \vdots & & \vdots & \vdots \\ X_1 & X_2 & \cdots & A_{n-1}^\complement & X_n \\ X_1 & X_2 & \cdots & X_{n-1} & A_n^\complement \end{array}\right ].\)

The diagonal components of this matrix \(A_i^\complement\) are equal correspondingly to \(X_i \smallsetminus A_i\).

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:

\(A^\complement = \left] A_1^\complement \quad A_2^\complement \quad \dots \quad A_n^\complement \right[.\)

\(R_1 = \left [\begin{array} {cccc} A_1 & A_2 & \dots & A_n\\ B_1 & B_2 & \dots & B_n \end{array}\right ],\)

\(R_1^\complement = \left ]\begin{array} {cccc} A_1^\complement & A_2^\complement & \dots & A_n^\complement \\ B_1^\complement & B_2^\complement & \dots & B_n^\complement \end{array}\right [.\)

Condensed: the full section is in Wikipedia.

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 \(\{X_i\}_{i\in I}\) is defined to be \[\prod_{i \in I} X_i = \left\{\left. f: I \to \bigcup_{i \in I} X_i\ \right|\ \forall i\in I.\ f(i) \in X_i\right\},\] 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. \(\prod_{i\in I} X_i\) may also be denoted \(\underset{i\in I} \operatorname{\mathsf{X}} X_i\).

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 j-th projection map.

Cartesian power is a Cartesian product where all the factors X_i 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 X. 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 i-th term in its corresponding set X_i. 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 \(\mathbb{R}^{\mathbb{N}}\).

If several sets are being multiplied together (e.g., X₁, X₂, X₃, ...), then some authors 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 \[(f\times g)(x, y) = (f(x), g(y)).\]

This can be extended to tuples and infinite collections of functions. This is different from the standard Cartesian product of functions considered as sets.