Union of Intersections

From ProofWiki
Jump to navigation Jump to search

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:

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