Union Distributes over Intersection/Examples/Arbitrary Integer Sets 2

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Example of Use of Union Distributes over Intersection

Let:

\(\ds A\) \(=\) \(\ds \set {2, 4, 6, 8, \dotsc}\)
\(\ds B\) \(=\) \(\ds \set {1, 3, 5, 7, \dotsc}\)
\(\ds C\) \(=\) \(\ds \set {1, 2, 3, 4}\)

Then:

$B \cup \paren {A \cap C} = \set {1, 2, 3, 4, 5, 7, \dotsc} = \paren {B \cup A} \cap \paren {B \cup C}$


Proof

\(\ds B \cup \paren {A \cap C}\) \(=\) \(\ds \set {1, 3, 5, 7, \dotsc} \cup \paren {\set {2, 4, 6, 8, \dotsc} \cap \set {1, 2, 3, 4} }\)
\(\ds \) \(=\) \(\ds \set {1, 3, 5, 7, \dotsc} \cup \set {2, 4}\)
\(\ds \) \(=\) \(\ds \set {1, 2, 3, 4, 6, 8, \dotsc}\)


\(\ds \paren {B \cup A} \cap \paren {B \cup C}\) \(=\) \(\ds \paren {\set {1, 3, 5, 7, \dotsc} \cup \set {2, 4, 6, 8, \dotsc} } \cap \paren {\set {1, 3, 5, 7, \dotsc} \cup \set {1, 2, 3, 4} }\)
\(\ds \) \(=\) \(\ds \set {1, 2, 3, 4, 5, 6, 7, 8, \dotsc} \cap \set {1, 2, 3, 4, 5, 7, \dotsc}\)
\(\ds \) \(=\) \(\ds \set {1, 2, 3, 4, 5, 7, \dotsc}\)

$\blacksquare$


Sources