Sets:Operations

Definitions

The two main set operations that we deal with are union and intersection. The union of two sets $A$ and $B$ is defined as $A\cup B=\{x:x\in A$ or $x\in B\}$ . For example:

The union of $A=\{1,3,5,7,9\}$ and $B=\{0,1,2,4,6,8,9\}$ is $A\cup B=\{0,1,2,3,4,5,6,7,8,9\}$
The union of the even integers and odd integers is $\mathbb {Z}$ .
The union of the set of rational numbers and the set of irrational numbers is $\mathbb {R}$ .
$A\cup A=A$ , and $\emptyset \cup A=A$ .
For sets $A$ and $B$ such that $A\subseteq B$ , $A\cup B=B$ , since all elements of $A$ are already in $B$ if $A\subseteq B$ .

The intersection of $A$ and $B$ is defined as $A\cap B=\{x:x\in A$ and $x\in B\}$ ; that is, the intersection of $A$ and $B$ is the set of all elements shared by the two sets. For example:

The intersection of $A=\{1,3,5,7,9\}$ and $B=\{0,1,2,4,6,8,9\}$ is $A\cap B=\{1,9\}$ .
The intersection of the even integers and odd integers is the empty set, since no element between these two sets is shared (an integer cannot be both even and odd).
$A\cap A=A$ , and $\emptyset \cap A=\emptyset$ .
For sets $A$ and $B$ such that $A\subseteq B$ , $A\cup B=A$ .