Mathematics-Online lexicon: Cartesian product

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	Mathematics-Online lexicon:
	Cartesian product

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

The cartesian product of two sets

and

is the set of all ordered pairs of elements of

and

$\displaystyle A\times B = \{(a,b):\ a\in A\land b\in B\} \,.$

Note that

$\displaystyle (a,b) = (a',b') \Leftrightarrow (a=a' \land b=b') \,.$

Hence (contrary to the identity of sets: $\{a,b\}=\{b,a\}$ ), the order of the elements is crucial.

Analogously, the -fold cartesian product

$\displaystyle A_1\times \cdots \times A_n$

is defined as the set of all ordered tupels $(a_1,\ldots,a_n)$ with $a_i\in A_i$ . If

for all $1 \leq i,j \leq n$ , then the notation $A^n = A\times \cdots \times A$ is used.

(Authors: Höllig/Hörner/Abele)

automatically generated 5/25/2009