Intersection Distributes over Union/Examples

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:

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