Cartesian product
For any sets $A$ and $B$, the Cartesian product $A\times B$ is the set consisting of all ordered pairs $(a,b)$ where $a\in A$ and $b\in B$.
The Cartesian product satisfies the following properties, for all sets $A$, $B$, $C$, and $D$:

•
$A\times \mathrm{\varnothing}=\mathrm{\varnothing}$

•
$(A\times B)\cap (C\times D)=(A\cap C)\times (B\cap D)$

•
${(A\times B)}^{\mathrm{\complement}}=({A}^{\mathrm{\complement}}\times {B}^{\mathrm{\complement}})\cup ({A}^{\mathrm{\complement}}\times B)\cup (A\times {B}^{\mathrm{\complement}})$
Here $\mathrm{\varnothing}$ denotes the empty set^{}, $\cap $ denotes intersection^{}, $\cup $ denotes union, and ${}^{\mathrm{\complement}}$ denotes complement with respect to some universal set $U$ containing $A$ and $B$.
Title  Cartesian product 

Canonical name  CartesianProduct 
Date of creation  20130322 11:48:56 
Last modified on  20130322 11:48:56 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  10 
Author  djao (24) 
Entry type  Definition 
Classification  msc 0300 
Classification  msc 81P10 
Classification  msc 81P05 
Related topic  GeneralizedCartesianProduct 