Union of Intersections

Theorem

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

Proof

\(\ds \paren {S_1 \cap S_2} \cup \paren {T_1 \cap T_2}\)

\(=\)

\(\ds \paren {\paren {S_1 \cap S_2} \cup T_1} \cap \paren {\paren {S_1 \cap S_2} \cup T_2}\)

Union Distributes over Intersection

\(\ds \)

\(=\)

\(\ds \paren {S_1 \cup T_1} \cap \paren {S_2 \cup T_1} \cap \paren {\paren {S_1 \cap S_2} \cup T_2}\)

Union Distributes over Intersection

\(\ds \)

\(\subseteq\)

\(\ds S_1 \cup T_1\)

Intersection is Subset

$\blacksquare$

Examples

Example: $4$ Arbitrarily Chosen Sets of Complex Numbers

Let:

\(\ds A\)

\(=\)

\(\ds \set {1, i, -i}\)

\(\ds B\)

\(=\)

\(\ds \set {2, 1, -i}\)

\(\ds C\)

\(=\)

\(\ds \set {i, -1, 1 + i}\)

\(\ds D\)

\(=\)

\(\ds \set {0, -i, 1}\)

Then:

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