Cartesian product

Last updated

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

Contents

[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

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, 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 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 , with denoting the real numbers) is thus assigned to the set of all points in the plane. [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 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,

[4]

because the ordered pairs are reversed unless at least one of the following conditions is satisfied: [8]

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): [8]

where denotes the absolute complement of A.

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 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: [1] 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 , can be defined as

An example of this is R3 = R × R × R, with R again the set of real numbers, [1] 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 I 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. may also be denoted . [11]

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 [12] choose to abbreviate the Cartesian product as simply ×Xi.

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, 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 set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.

In mathematics, one can often define a direct product of objects already known, giving a new one. This induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. More abstractly, one talks about the product in category theory, which formalizes these notions.

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize the concept of a sequence in a metric space. Nets are primarily used in the fields of Analysis and Topology, where they are used to characterize many important topological properties that, sequences are unable to characterize. Nets are in one-to-one correspondence with filters.

In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

In mathematics, a base (or basis; pl.: bases) for the topology τ of a topological space (X, τ) is a family of open subsets of X such that every open set of the topology is equal to the union of some sub-family of . For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.

In mathematics, a tuple is a finite sequence or ordered list of numbers or, more generally, mathematical objects, which are called the elements of the tuple. An n-tuple is a tuple of n elements, where n is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively.

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

<span class="mw-page-title-main">Projective variety</span>

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .

In general topology, a branch of mathematics, a non-empty family A of subsets of a set is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of is infinite. Sets with the finite intersection property are also called centered systems and filter subbases.

In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.

In topology, the cartesian product of topological spaces can be given several different topologies. One of the more natural choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces. Another possibility is the product topology, where a base is given by the Cartesian products of open sets in the component spaces, only finitely many of which can be not equal to the entire component space.

<span class="mw-page-title-main">Three-dimensional space</span> Geometric model of the physical space

In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, embedding problem, information theory, and other areas.

In mathematics, a filter on a set is a family of subsets such that:

  1. and
  2. if and , then
  3. If , and , then

In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, where a given function selects one real number for each integer as indexing.

<span class="mw-page-title-main">Oriented matroid</span> Abstraction of ordered linear algebra

An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over ordered fields. In comparison, an ordinary matroid abstracts the dependence properties that are common both to graphs, which are not necessarily directed, and to arrangements of vectors over fields, which are not necessarily ordered.

In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalizations in the setting of symplectic geometry. A few important generalizations are Kostant's convexity theorem, Atiyah–Guillemin–Sternberg convexity theorem, Kirwan convexity theorem.

In mathematical analysis and 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.

<span class="mw-page-title-main">Filters in topology</span> Use of filters to describe and characterize all basic topological notions and results.

Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such as convergence, continuity, compactness, and more. Filters, which are special families of subsets of some given set, 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.

References

  1. 1 2 3 Weisstein, Eric W. "Cartesian Product". MathWorld . Retrieved September 5, 2020.
  2. Warner, S. (1990). Modern Algebra. Dover Publications. p. 6.
  3. Nykamp, Duane. "Cartesian product definition". Math Insight. Retrieved September 5, 2020.
  4. 1 2 3 "Cartesian Product". web.mnstate.edu. Archived from the original on July 18, 2020. Retrieved September 5, 2020.
  5. "Cartesian". Merriam-Webster.com. 2009. Retrieved December 1, 2009.
  6. Corry, S. "A Sketch of the Rudiments of Set Theory" (PDF). Retrieved May 5, 2023.
  7. Goldberg, Samuel (1986). Probability: An Introduction. Dover Books on Mathematics. Courier Corporation. p. 41. ISBN   9780486652528.
  8. 1 2 Singh, S. (August 27, 2009). Cartesian product. Retrieved from the Connexions Web site: http://cnx.org/content/m15207/1.5/
  9. 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
  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 http://www.mathpath.org/concepts/infinity.htm
  11. 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.
  12. Osborne, M., and Rubinstein, A., 1994. A Course in Game Theory. MIT Press.