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# Cartesian product  ( Axiom Of Choice )

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In (and, usually, in other parts of ), 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 is the set of all where and . Products can be specified using set-builder notation, e.g.

$A\times B = \\left\{\,\left(a,b\right)\mid a\in A \ \mbox\left\{ and \right\} \ b\in B\,\\right\}.$

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

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-. An ordered pair is a 2-tuple or couple.

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 .

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 , which correspond to all 52 possible playing cards.

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


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 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 , called its coordinates. Usually, such a pair's first and second components are called its x and y coordinates, respectively; cf. picture. The set of all such pairs (i.e. the Cartesian product 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 principles follows from a definition of . The most common definition of ordered pairs, the Kuratowski definition, is $\left(x, y\right) = \\left\{\\left\{x\\right\},\\left\{x, y\\right\}\\right\}$. Under this definition, $\left(x, y\right)$ is an element of $\mathcal\left\{P\right\}\left(\mathcal\left\{P\right\}\left(X \cup Y\right)\right)$, and $X\times Y$ is a subset of that set, where $\mathcal\left\{P\right\}$ represents the operator. Therefore, the existence of the Cartesian product of any two sets in 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 is not ,

$A \times B \neq B \times A,$
because the are reversed unless at least one of the following conditions is satisfied:
• A is equal to B, or
• A or B is the .

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 (unless one of the involved sets is empty).

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

Intersections, unions, and subsets
 Example sets ={ y∈ℝ:1≤ y≤4}, ={ x∈ℝ:2≤ x≤5}, and ={ x∈ℝ:4≤ x≤7}, demonstrating A×( B∩ C) = ()∩(), A×( B∪ C) = ()∪(), and A×( B\ C) = ()\().]]
 Example sets ={ x∈ℝ:2≤ x≤5}, ={ x∈ℝ:3≤ x≤7}, ={ y∈ℝ:1≤ y≤3}, ={ y∈ℝ:2≤ y≤4}, demonstrating = .]] 
The Cartesian product behaves nicely with respect to intersections, cf. left picture.
$\left(A \cap B\right) \times \left(C \cap D\right) = \left(A \times C\right) \cap \left(B \times D\right)$

In most cases the above statement is not true if we replace intersection with union, cf. middle picture.

$\left(A \cup B\right) \times \left(C \cup D\right) \neq \left(A \times C\right) \cup \left(B \times D\right)$

In fact, we have that:

$\left(A \times C\right) \cup \left(B \times D\right) = \left(A \cup \left(A \cup \left(B$

For the set difference we also have the following identity:

$\left(A \times C\right) \setminus \left(B \times D\right) = A \cup \left(A$

Here are some rules demonstrating distributivity with other operators (cf. right picture):Singh, S. (August 27, 2009). Cartesian product. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/

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

Other properties related with are:

$\text\left\{if \right\} A \subseteq B \text\left\{ then \right\} A \times C \subseteq B \times C,$
$\text\left\{if both \right\} A,B \neq \emptyset \text\left\{ then \right\} A \times B \subseteq C \times D \iff A \subseteq C\text\left\{ and \right\} B \subseteq D.$Cartesian Product of Subsets. (February 15, 2011). ProofWiki /ref>

Cardinality
The of a set is the number of elements of the set. For example, defining two sets: } and Both set A and set B consist of two elements each. Their Cartesian product, written as , 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 is if either A or B is infinite and the other set is not the empty set.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

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

$X_1\times\cdots\times X_n = \\left\{\left(x_1, \ldots, x_n\right) \mid x_i \in X_i \ \text\left\{for every\right\} \ i \in \\left\{1, \ldots, n\\right\} \\right\}.$

of . If tuples are defined as nested ordered pairs, it can be identified with . If a tuple is defined as a function on 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

$\\left\{ x:\\left\{1,\ldots,n\\right\}\to X_1\cup\ldots\cup X_n \ | \ x\left(i\right)\in X_i \ \text\left\{for every\right\} \ i \in \\left\{1, \ldots, n\\right\} \\right\}.$

n-ary Cartesian power
The Cartesian square of a set X is the Cartesian product . An example is the 2-dimensional plane where R is the set of : R2 is the set of all points 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

$X^n = \underbrace\left\{ X \times X \times \cdots \times X \right\}_\left\{n\right\}= \\left\{ \left(x_1,\ldots,x_n\right) \ | \ x_i \in X \ \text\left\{for every\right\} \ i \in \\left\{1, \ldots, n\\right\} \\right\}.$

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

The n-ary cartesian power of a set X is 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 , corresponding to the with X.

Infinite Cartesian products
It is possible to define the Cartesian product of an arbitrary (possibly ) of sets. If I is any , and $\\left\{X_i\\right\}_\left\{i\in I\right\}$ is a family of sets indexed by I, then the Cartesian product of the sets in X is defined to be

$\prod_\left\{i \in I\right\} X_i = \left\\left\{\left. f: I \to \bigcup_\left\{i \in I\right\} X_i\ \right|\ \left(\forall i\right)\left(f\left(i\right) \in X_i\right)\right\\right\},$

that is, the set of all functions defined on the 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

$\pi_\left\{j\right\}: \prod_\left\{i \in I\right\} X_i \to X_\left\{j\right\},$
defined by $\pi_\left\{j\right\}\left(f\right) = f\left(j\right)$ 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,

$\prod_\left\{i \in I\right\} X_i = \prod_\left\{i \in I\right\} X$
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 $\mathbb\left\{N\right\}$, the : this Cartesian product is the set of all infinite sequences with the ith term in its corresponding set Xi. For example, each element of
$\prod_\left\{n = 1\right\}^\infty \mathbb R = \mathbb R \times \mathbb R \times \cdots$
can be visualized as a with countably infinite real number components. This set is frequently denoted $\mathbb\left\{R\right\}^\omega$, or $\mathbb\left\{R\right\}^\left\{\mathbb\left\{N\right\}\right\}$.

Other forms

Abbreviated form
If several sets are being multiplied together, e.g. X1, X2, X3, …, then some authorsOsborne, M., and Rubinstein, A., 1994. A Course in Game Theory. MIT Press. 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 is a function from to with
$\left(f\times g\right)\left(a, x\right) = \left(f\left(a\right), g\left(x\right)\right).$

This can be extended to 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 $\mathbb\left\{N\right\}$, then the cylinder of $B$ is $B \times \mathbb\left\{N\right\}$.

Definitions outside set theory

Category theory
Although the Cartesian product is traditionally applied to sets, 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 .

Exponentiation is the of the Cartesian product; thus any category with a Cartesian product (and a ) is a Cartesian closed category.

Graph theory
In the Cartesian product of two graphs G and H is the graph denoted by whose vertex set is the (ordinary) Cartesian product and such that two vertices ( u, v) and ( u′, v′) are adjacent in if and only if and v is adjacent with v′ in H, or 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.

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