Definition:Cartesian Product

Definition

The cartesian product (or Cartesian product) of two sets $S$ and $T$ is the set of ordered pairs $\left({x, y}\right)$ with $x \in S$ and $y \in T$.

This is denoted:

$S \times T = \left\{{\left({x, y}\right) : x \in S \land y \in T}\right\}$

Some authors call this the direct product of $S$ and $T$.

Some call it the cartesian product set, others just the product set.

Another way of defining it is by:

$\left({x, y}\right) \in S \times T \iff x \in S, y \in T$

It is also known as the cross product of two sets, but this can be confused with other usages of this term.

Factors

In a cartesian product $S \times T$, the sets $S$ and $T$ are called the factors of $S \times T$.

General Definition

Let $\left \langle {S_n} \right \rangle$ be a sequence of sets.

The cartesian product of $\left \langle {S_n} \right \rangle$ is defined as:

$\displaystyle \prod_{k=1}^n S_k = \left\{{\left({x_1, x_2, \ldots, x_n}\right): \forall k \in \N^*_n: x_k \in S_k}\right\}$

It is also denoted $S_1 \times S_2 \times \ldots \times S_n$.

Thus $S_1 \times S_2 \times \ldots \times S_n$ is the set of all ordered $n$-tuples $\left({x_1, x_2, \ldots, x_n}\right)$ with $x_k \in S_k$.

Cartesian Space

Let $S$ be a set.

Then the cartesian $n$th power of $S$, or $S$ to the power of $n$, is defined as:

$\displaystyle S^n = \prod_{k=1}^n S = \left\{{\left({x_1, x_2, \ldots, x_n}\right): \forall k \in \N^*_n: x_k \in S}\right\}$

Thus $S^n = S \times S \times \ldots \left({n}\right) \ldots \times S$

Alternatively it can be defined recursively:

$S^n = \begin{cases} S: & n = 1 \\ S \times S^{n-1} & n > 1 \end{cases}$

The set $S^n$ called a cartesian space.

An element $x_j$ of a tuple $\left({x_1, x_2, \ldots, x_n}\right)$ of a cartesian space $S^n$ is known as a basis element of $S^n$.

Real Cartesian Space

When $S$ is the set of real numbers $\R$, the cartesian product takes on a special significance.

Let $n \in \N^*$.

Then $\R^n$ is the cartesian product defined as follows:

$\displaystyle \R^n = \R \times \R \times \cdots \left({n}\right) \cdots \times \R = \prod_{k=1}^n \R$

Similarly, $\R^n$ can be defined as the set of all real $n$-tuples:

$\R^n = \left\{{\left({x_1, x_2, \ldots, x_n}\right): x_1, x_2, \ldots, x_n \in \R}\right\}$

It can be shown that:

• $\R^2$ is isomorphic to any infinite flat plane in space;
• $\R^3$ is isomorphic to the whole of space itself.

See the definition of a Real Vector Space.

Axiomatic Set Theory

The concept of the cartesian product is shown in Kuratowski Formalization of Ordered Pair to be constructible from the Zermelo-Fraenkel axioms.

Notes

The notation for the cartesian power of a set $S^n$ should not be confused with the notation used for the conjugate of a set.

Also beware not to confuse the name of the concept itself with that of the power set $\mathcal P \left({S}\right)$ of $S$.

Also see

• Cartesian products of algebraic structures:
  • External Direct Product
  • Internal Direct Product
  • (External) Group Direct Product
  • Internal Group Direct Product

• Results about Cartesian products can be found here.

Source of Name

This entry was named for René Descartes.

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 2$
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 6$: Ordered Pairs
• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.7$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.2$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Introduction
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.1$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 9$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 3$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 8$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.8$