Operations on Sets
Recall that a set is a collection of elements.
Given sets $A$ and $B$ , we can define the following operations:
Operation

Notation

Meaning

Intersection

$A\cap B$

all elements which are in both
$A$
and
$B$

Union

$A\cup B$

all elements which are in either
$A$
or
$B$
(or both)

Difference

$AB$

all elements which are in
$A$
but not in
$B$

Complement

$\stackrel{\xaf}{A}$
(or
${A}^{C}$
)

all elements which are not in
$A$

Example 1:
Let $A=\left\{1,2,3,4\right\}$ and let $B=\left\{3,4,5,6\right\}$ .
Then:
$A\cap B=\left\{3,4\right\}$
$A\cup B=\left\{1,2,3,4,5,6\right\}$
$AB=\left\{1,2\right\}$
${A}^{C}=\left\{\text{allrealnumbersexcept}1,2,3\text{and}4\right\}$
Example 2:
Let $A=\left\{y,z\right\}$ and let $B=\left\{x,y,z\right\}$ .
Then:
$\begin{array}{l}A\cap B=\left\{y,z\right\}\\ A\cup B=\left\{x,y,z\right\}\\ AB=\varnothing \\ {A}^{C}=\left\{\text{everythingexcept}y\text{and}z\right\}\end{array}$