Cartesian product

Cartesian square redirects here. For Cartesian squares in category theory, see Cartesian square (category theory). In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, a branch of Mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a

In mathematics, the Cartesian product (or product set) is a direct product of sets. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, one can often define a direct product of objectsalready known giving a new one The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry

Specifically, the Cartesian product of two sets X (for example the points on an x-axis) and Y (for example the points on a y-axis), denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y (e. In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry g. the whole of the x-y plane):

$X\times Y = \{(x,y) | x\in X\;\mathrm{and}\;y\in Y\}.$

For example, the Cartesian product of the thirteen-element set of standard playing card ranks {Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2} and the four-element set of card suits {♠, ♥, ♦, ♣} is the 52-element set of playing cards {(Ace, ♠), (King, ♠), . . . , (2, ♠), (Ace, ♥), . . . , (3, ♣), (2, ♣)}. The Cartesian product has 52 elements because that is the product of 13 times 4.

A Cartesian product of two finite sets can be represented by a table, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table, by choosing the element of the set from the row and the column.

1 n-ary product
2 Cartesian square and Cartesian power
3 Infinite products
4 Abbreviated form
5 Cartesian product of functions
6 Category theory
7 See also
8 External links
9 References

n-ary product

The Cartesian product can be generalized to the n-ary Cartesian product over n sets X₁, . . . , X_n:

$X_1\times\cdots\times X_n = \{(x_1, \ldots, x_n) \mid x_1\in X_1\;\mathrm{and}\;\cdots\;\mathrm{and}\;x_n\in X_n\}.$

Indeed, it can be identified to (X₁ × . . . × X_n-1) × X_n. It is a set of n-tuples. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple

Cartesian square and Cartesian power

The Cartesian square (or binary Cartesian product) 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 - all points (x,y) where x and y are real numbers (see the Cartesian coordinate system). In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

The cartesian power of a set X can be defined as:

$X^n = \underbrace{ X \times X \cdots\times X }_{n}= \left\{ (x_1, x_2, \ldots, x_n) \ | \ x_1 \in X \land x_2 \in X \land \ldots x_n \in X \right\}.$

An example of this is R³ = R × R × R, with R again the set of real numbers, and more generally Rⁿ.

Infinite products

It is possible to define the Cartesian product of an arbitrary (possibly infinite) family of sets. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In Set theory and related branches of Mathematics, a collection F of Subsets of a given set S is called a family of subsets of S If I is any index set, and {X_i | i ∈ I} is a collection of sets indexed by I, then the Cartesian product of the sets in X is defined to be

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

that is, the set of all functions defined on the index set such that the value of the function at a particular index i is an element of X_i . In Mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index In Mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index

For each j in I, the function

$\pi_{j} : \prod_{i \in I} X_i \to X_{j},$

defined by π_j(f) = f(j) is called the j -th projection map.

An important case is when the index set is 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. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an For example, each element of

$\prod_{n = 1}^\infty \mathbb R =\mathbb{R}^\omega= \mathbb R \times \mathbb R \times \cdots,$

can be visualized as a vector with an infinite number of real-number components.

The special case Cartesian exponentiation occurs when all the factors X_i involved in the product 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. This case is important in the study of cardinal exponentiation. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.

The definition of finite Cartesian products can be seen as a special case of the definition for infinite products. In this interpretation, an n-tuple can be viewed as a function on {1, 2, . . . , n} that takes its value at i to be the i-th element of the tuple (in some settings, this is taken as the very definition of an n-tuple).

Nothing in the definition of infinite Cartesian products implies that the Cartesian product of nonempty sets must itself be nonempty. This assertion is equivalent to the axiom of choice. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

Abbreviated form

If several sets are being multiplied together, e. g. X₁, X₂, X₃, …, then some authors^[1] 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 f×g is a function from A×X to B×Y with

$(f\times g)(a, x) = (f(a), g(x))$

As above this can be extended to tuples and infinite collections of functions. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple

Category theory

Although the Cartesian product is traditionally applied to sets, category theory provides a more general interpretation of the product of mathematical structures. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as

External links

Cartesian Product at ProvenMath

References

^ M. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of In Mathematics, an empty product, or nullary product, is the result of multiplying no numbers In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations An ultraproduct is a mathematical construction of which the ultrapower (defined below is a special case Osborne and A. Rubinstein, A Course in Game Theory, MIT Press 1994.

Dictionary

-noun

(set theory) The set of all possible pairs of elements whose components are members of two sets. Notation: <math>X \times Y = \{(x,y)\|x\in X \land y\in Y\}</math>.

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