set difference
Definition
Let $A$ and $B$ be sets. The set difference^{} (or simply difference) between $A$ and $B$ (in that order) is the set of all elements of $A$ that are not in $B$. This set is denoted by $A\setminus B$ or $AB$ (or occasionally $A\sim B$). So we have
$$A\setminus B=\{x\in A\mid x\notin B\}.$$ 
Venn diagram^{} showing $A\setminus B$ in blue 
Properties
Here are some properties of the set difference operation^{}:

1.
If $A$ is a set, then
$$A\setminus \mathrm{\varnothing}=A$$ and
$$A\setminus A=\mathrm{\varnothing}=\mathrm{\varnothing}\setminus A.$$ 
2.
If $A$ and $B$ are sets, then
$$B\setminus (A\cap B)=B\setminus A.$$ 
3.
If $A$ and $B$ are subsets of a set $X$, then
$$A\setminus B=A\cap {B}^{\mathrm{\complement}}$$ and
$${(A\setminus B)}^{\mathrm{\complement}}={A}^{\mathrm{\complement}}\cup B,$$ where ${}^{\mathrm{\complement}}$ denotes complement in $X$.

4.
If $A$, $B$, $C$ and $D$ are sets, then
$$(A\setminus B)\cap (C\setminus D)=(A\cap C)\setminus (B\cup D).$$
Remark
As noted above, the set difference is sometimes written as $AB$. However, if $A$ and $B$ are sets in a vector space (or, more generally, a module (http://planetmath.org/Module)), then $AB$ is commonly used to denote the set
$$AB=\{ab\mid a\in A,b\in B\}$$ 
rather than the set difference.
Title  set difference 

Canonical name  SetDifference 
Date of creation  20130322 11:59:38 
Last modified on  20130322 11:59:38 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  33 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 03E20 
Synonym  difference between sets 
Synonym  difference 
Related topic  SymmetricDifference 
Related topic  InverseImageCommutesWithSetOperations 
Related topic  Difference2 