# Set Intersection/Examples/4 Arbitrarily Chosen Sets of Complex Numbers

## Example of Set Intersection

Let:

 $\displaystyle A$ $=$ $\displaystyle \set {1, i, -i}$ $\displaystyle B$ $=$ $\displaystyle \set {2, 1, -i}$ $\displaystyle C$ $=$ $\displaystyle \set {i, -1, 1 + i}$ $\displaystyle D$ $=$ $\displaystyle \set {0, -i, 1}$

Then:

$\paren {A \cup C} \cap \paren {B \cup D} = \set {1, -i}$

## Proof

 $\displaystyle \paren {A \cup C} \cap \paren {B \cup D}$ $=$ $\displaystyle \paren {\set {1, i, -i} \cup \set {i, -1, 1 + i} } \cap \paren {\set {2, 1, -i} \cup \set {0, -i, 1} }$ $\displaystyle$ $=$ $\displaystyle \set {1, i, -i, -1, 1 + i} \cap \set {2, 1, -i, 0}$ Definition of Set Union $\displaystyle$ $=$ $\displaystyle \set {1, -i}$ Definition of Set Intersection

$\blacksquare$