Union of Intersections/Examples/4 Arbitrarily Chosen Sets of Complex Numbers

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Example of Union of Intersections

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:

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


Proof

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

$\blacksquare$


Sources