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 $A-B$(or occasionally $A\sim B$). So we have

$$A\setminus B=\{x\in A\mid x\notin B\}.$$

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 $A-B$.However, if $A$ and $B$ are sets in a vector space(or, more generally, a module (http://planetmath.org/Module)),then $A-B$ is commonly used to denote the set

$$A-B=\{a-b\mid a\in A,b\in B\}$$

rather than the set difference.