Set Intersection/Examples/4 Arbitrarily Chosen Sets of Complex Numbers

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Example of Set Intersection

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:

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


Proof

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

$\blacksquare$


Sources