# Set Differences

# Why

We consider elements of one set which are not
contained in another set.

# Definition

Let $A$ and $B$ denote sets.
The difference between
$A$ and $B$ is the set $\Set{x \in A}{x
\not\in B}$.
In other words, the difference between $A$ and
$B$ is the set of all points of $A$ which do
not belong to $B$.

It is not necessary that $B \subset A$; the
difference is called
proper if $A \supset B$.
This terminology is from that of proper subsets.

## Notation

We denote the difference between $A$ and $B$
by $A \setminus B$.
Other notations used include $-$ or $\sim$.

# Properties

The following are straightforward.

$A \setminus \varnothing = A$

$A \setminus A = \varnothing$