set operations

We can use set operations to transform a set or a combination (a set?) of sets.

Set Complement

The complement of a set $A$ is equal to the set of elements from a universe set $A$ not belonging to set $A$, denoted $\overline{A}=\{x\in U:x\notin A\}$.

Set Difference

The different of two sets $A$ and $B$ is the set of elements in A but not in B, which is denoted $A-B=A\setminus B=A\cap \overline{B}=\{x:x\in A\text{ and }x\notin B\}$.

Set Union

The union of two sets $A$ and $B$ is the set containing elements from both $A$ and $B$. For an extended usage of set union, use the set union symbol like the sigma symbol for sums.

Set Intersection

The intersection of two sets $A$ and $B$ is the set containing elements that are found in both $A$ and $B$. For an extended usage of set intersection, use the set intersection symbol like the sigma symbol for sums.

Cross Product

The cross product of set $A$ and $B$ is the set containing all possible ordered pairs whose first component is an element of $A$ and whose second component is an element of $B$. If $A$ and $B$ are finite sets, then $|A\times B|=|A|\cdot |B|$.