# Sets/Operations/Introduction/Section

Similar to the connection of statements to get new statements, there are operations to make new sets from old ones. The most important operations are the following.[1]

1. Union
${\displaystyle {}A\cup B:={\left\{x\mid x\in A{\text{ or }}x\in B\right\}}\,.}$
2. Intersection
${\displaystyle {}A\cap B:={\left\{x\mid x\in A{\text{ and }}x\in B\right\}}\,.}$
3. Difference set
${\displaystyle {}A\setminus B:={\left\{x\mid x\in A{\text{ and }}x\notin B\right\}}\,.}$

For these operations to make sense, the sets need to be subsets of a common basic set. This ensures that we are talking about the same elements. Quite often this basic set is not mentioned explicitly and has to be understood from the context. A special case of the difference set is the complement of a subset ${\displaystyle {}A\subseteq G}$ in a given base set ${\displaystyle {}G}$, also denoted as

${\displaystyle {}\complement A:=G\setminus A={\left\{x\in G\mid x\not \in A\right\}}\,.}$

If two sets have an empty intersection, meaning ${\displaystyle {}A\cap B=\emptyset }$, we also say that they are disjoint.

1. It is easy to memorize the symbols: the ${\displaystyle {}\cup }$ for union looks like u. The intersection is written as ${\displaystyle {}\cap }$. The corresponding logical operations or, and have the analog form ${\displaystyle {}\vee }$ and ${\displaystyle {}\wedge }$ respectively.