Sets/Operations/Introduction/Section

Similar to the construction of new statements from given statements by connecting them with logical connectives, there are operations to make new sets from old ones. The most important operations are the following:

1. Union

   ${}A\cup B:={\left\{x\mid x\in A{\text{ or }}x\in B\right\}}\,.$

2. Intersection

   ${}A\cap B:={\left\{x\mid x\in A{\text{ and }}x\in B\right\}}\,.$

3. Difference set

   ${}A\setminus B:={\left\{x\mid x\in A{\text{ and }}x\notin B\right\}}\,.$

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 ${}A\subseteq G$ in a given set ${}G$, also denoted as

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

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