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. 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 base 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.