Intersection Distributes over Union/Examples
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Examples of Use of Intersection Distributes over Union
$3$ Arbitrarily Chosen Sets
Let:
\(\ds A\) | \(=\) | \(\ds \set {3, -i, 4, 2 + i, 5}\) | ||||||||||||
\(\ds B\) | \(=\) | \(\ds \set {-i, 0, -1, 2 + i}\) | ||||||||||||
\(\ds C\) | \(=\) | \(\ds \set {- \sqrt 2 i, \dfrac 1 2, 3}\) |
Intersection with Union
\(\ds A \cap \paren {B \cup C}\) | \(=\) | \(\ds \set {3, -i, 4, 2 + i, 5} \cap \set {-i, 0, -\sqrt 2 i, -1, 2 + i, \dfrac 1 2, 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {3, -i, 2 + i}\) |
Union of Intersections
\(\ds \paren {A \cap B} \cup \paren {A \cap C}\) | \(=\) | \(\ds \set {-i, 2 + i} \cup \set 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {3, -i, 2 + i}\) |
Thus it is seen that:
- $A \cap \paren {B \cup C} = \paren {A \cap B} \cup \paren {A \cap C}$
Arbitrary Integer Sets
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:
- $\paren {A \cup B} \cap C = \set {1, 2, 3, 4} = \paren {A \cap C} \cup \paren {B \cap C}$