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

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

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 $A-B$. 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.

Title	set difference
Canonical name	SetDifference
Date of creation	2013-03-22 11:59:38
Last modified on	2013-03-22 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