# Union of Intersections

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## Theorem

$\paren {S_1 \cap S_2} \cup \paren {T_1 \cap T_2} \subseteq S_1 \cup T_1$

## Proof

 $\displaystyle \paren {S_1 \cap S_2} \cup \paren {T_1 \cap T_2}$ $=$ $\displaystyle \paren {\paren {S_1 \cap S_2} \cup T_1} \cap \paren {\paren {S_1 \cap S_2} \cup T_2}$ Union Distributes over Intersection $\displaystyle$ $=$ $\displaystyle \paren {S_1 \cup T_1} \cap \paren {S_2 \cup T_1} \cap \paren {\paren {S_1 \cap S_2} \cup T_2}$ Union Distributes over Intersection $\displaystyle$ $\subseteq$ $\displaystyle S_1 \cup T_1$ Intersection is Subset

$\blacksquare$

## Examples

### Example: $4$ Arbitrarily Chosen Sets of Complex Numbers

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:

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