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set difference (Definition)

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$. So we have
$\displaystyle A\setminus B = \{ x\in A \mid x \notin B\}. $
\includegraphics[scale=1]{venn.eps}
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
    $\displaystyle A\setminus\varnothing = A $
    and
    $\displaystyle A\setminus A = \varnothing = \varnothing \setminus A. $
  2. If $ A$ and $ B$ are sets, then
    $\displaystyle B\setminus(A\cap B) = B\setminus A. $
  3. If $ A$ and $ B$ are subsets of a set $ X$, then
    $\displaystyle A\setminus B = A\cap B^\complement $
    and
    $\displaystyle (A\setminus B)^\complement = A^\complement \cup B, $
    where $ ^\complement$ denotes complement in $ X$.
  4. If $ A$, $ B$, $ C$ and $ D$ are sets, then
    $\displaystyle (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), then $ A-B$ is commonly used to denote the set
$\displaystyle 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.



"set difference" is owned by yark. [ full author list (3) | owner history (2) ]
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See Also: symmetric difference, the inverse image commutes with set operations, difference

Other names:  difference between sets, difference
Keywords:  set

Attachments:
properties of set difference (Derivation) by CWoo
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Cross-references: module, vector space, complement, subsets, Venn diagram
There are 23 references to this entry.

This is version 26 of set difference, born on 2001-11-16, modified 2008-03-25.
Object id is 913, canonical name is SetDifference.
Accessed 22208 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
Pending Errata and Addenda
None.
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Discussion
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forum policy
for example by akrowne on 2001-11-17 10:44:36
Often one will see \mathbb{Z} \ {0} to mean "integers except zero", etc
-apk
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