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'set difference'
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| Title of object: |
set difference |
| Canonical Name: |
SetDifference |
| Type: |
Definition |
| Created on: |
2001-11-16 19:51:43 |
| Modified on: |
2008-03-25 13:11:25 |
| Classification: |
msc:03E20 |
| Keywords: |
set |
| Synonyms: |
set difference=difference between sets
set difference=difference |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}
\def\emptyset{\varnothing} |
Content:
\PMlinkescapeword{between}
\PMlinkescapeword{operation}
\PMlinkescapeword{order}
\PMlinkescapeword{properties}
\section*{Definition}
Let $A$ and $B$ be sets.
The \emph{set difference} (or simply \emph{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 $A-B$.
So we have
\[
A\setminus B = \{ x\in A \mid x \notin B\}.
\]
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=1]{venn.eps} \\
{\tiny Venn diagram showing $A\setminus B$ in blue}
\end{tabular}
\end{center}
\section*{Properties}
Here are some properties of the set difference operation:
\begin{enumerate}
\item If $A$ is a set, then
\[
A\setminus\emptyset = A
\]
and
\[
A\setminus A = \emptyset = \emptyset\setminus A.
\]
\item If $A$ and $B$ are sets, then
\[
B\setminus(A\cap B) = B\setminus A.
\]
\item If $A$ and $B$ are subsets of a set $X$, then
\[
A\setminus B = A\cap B^\complement
\]
and
\[
(A\setminus B)^\complement = A^\complement \cup B,
\]
where $^\complement$ denotes complement in $X$.
\item If $A$, $B$, $C$ and $D$ are sets, then
\[
(A\setminus B)\cap (C\setminus D) = (A\cap C)\setminus (B\cup D).
\]
\end{enumerate}
\section*{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), then $A-B$ is commonly used to denote the set
\[
A-B = \{ a-b \mid a\in A, b\in B\},
\]
which is not usually the same as the set difference of $A$ and $B$.
Using the notation $A-B$ for set difference can therefore cause confusion,
and so is probably best avoided. |
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