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Mathematics-Online course: Basic Mathematics - Sets

Cartesian Product

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The cartesian product of two sets $ A$ and $ B$ is the set of all ordered pairs of elements of $ A$ and $ B$:

$\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 $ n$-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 $ A_i = A_j$ for all $ 1 \leq i,j \leq n$, then the notation $ A^n = A\times \cdots \times A$ is used.
(Authors: Höllig/Hörner/Abele)
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  automatically generated 10/31/2008