# 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 in A and b is in B.^{[1]} In terms of set-builder notation, that is
^{[2]}^{[3]}

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).^{[4]}

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,^{[5]} 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 and , with the typical Kuratowski's definition of a pair as , an appropriate domain is the set where denotes the power set. Then the Cartesian product of the sets and would be defined as^{[6]}

## Examples

### A deck of cards

An illustrative example is the

*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.

### A two-dimensional coordinate system

The main historical example is the

^{[7]}

## 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

### Non-commutativity and non-associativity

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

The Cartesian product *A* × *B* is not

^{}[4] because the ordered pairs are reversed unless at least one of the following conditions is satisfied:

^{[8]}

- 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

*A*= {1}, then (

*A*×

*A*) ×

*A*= {((1, 1), 1)} ≠ {(1, (1, 1))} =

*A*× (

*A*×

*A*).

### Intersections, unions, and subsets

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):^{[8]}
where denotes the

Other properties related with subsets are:

^{[9]}

### 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)}.

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*|.^{[4]}

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.^{[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 *X*_{1}, ..., *X*_{n} as the set

of n-tuples. If tuples are defined as nested ordered pairs, it can be identified with (*X*_{1} × ... × *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

*X*

_{1}× ... ×

*X*

_{n}is the set of functions

*n*-ary Cartesian power

The **Cartesian square** of a set X is the Cartesian product *X*^{2} = *X* × *X*.
An example is the 2-dimensional plane **R**^{2} = **R** × **R** where **R** is the set of real numbers:^{[1]} **R**^{2} 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 , can be defined as

An example of this is **R**^{3} = **R** × **R** × **R**, with **R** again the set of real numbers,^{[1]} and more generally **R**^{n}.

The n-ary Cartesian power of a set X is

### 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 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*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. may also be denoted .

_{i}^{[11]}

For each j in I, the function
defined by 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,
is the set of all functions from I to X, and is frequently denoted

*X*. This case is important in the study of

^{I}*X*. For example, each element of can be visualized as a vector with countably infinite real number components. This set is frequently denoted , or .

_{i}## Other forms

### Abbreviated form

If several sets are being multiplied together (e.g., *X*_{1}, *X*_{2}, *X*_{3}, ...), then some authors^{[12]} choose to abbreviate the Cartesian product as simply ×*X*_{i}.

### Cartesian product of functions

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

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 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,

### 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, *or* *v* = *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

- 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

## References

- ^
^{a}^{b}^{c}Weisstein, Eric W. "Cartesian Product".*MathWorld*. Retrieved September 5, 2020. **^**Warner, S. (1990).*Modern Algebra*. Dover Publications. p. 6.**^**Nykamp, Duane. "Cartesian product definition".*Math Insight*. Retrieved September 5, 2020.- ^
^{a}^{b}^{c}"Cartesian Product".*web.mnstate.edu*. Archived from the original on July 18, 2020. Retrieved September 5, 2020. **^**"Cartesian".*Merriam-Webster.com*. 2009. Retrieved December 1, 2009.**^**Corry, S. "A Sketch of the Rudiments of Set Theory" (PDF). Retrieved May 5, 2023.- ISBN 9780486652528.
- ^
^{a}^{b}Singh, S. (August 27, 2009).*Cartesian product*. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/ **^**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**^**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**^**F. R. Drake,*Set Theory: An Introduction to Large Cardinals*, p. 24. Studies in Logic and the Foundations of Mathematics, vol. 76 (1978). ISBN 0-7204-2200-0.**^**Osborne, M., and Rubinstein, A., 1994.*A Course in Game Theory*. MIT Press.

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