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
\(\ds \paren {S_1 \cap S_2} \cup \paren {T_1 \cap T_2}\) | \(=\) | \(\ds \paren {\paren {S_1 \cap S_2} \cup T_1} \cap \paren {\paren {S_1 \cap S_2} \cup T_2}\) | 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:
- $\paren {A \cap C} \cup \paren {B \cap D} = \set {1, i, -i}$