Intersection Distributes over Union/Examples/Arbitrary Integer Sets

Example of Use of Intersection Distributes over Union

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:

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

$\ds \paren {A \cup B} \cap C$

$=$

$\ds \paren {\set {2, 4, 6, 8, \dotsc} \cup \set {1, 3, 5, 7, \dotsc} } \cap \set {1, 2, 3, 4}$

$\ds $

$=$

$\ds \set {1, 2, 3, 4, 5, 6, 7, 8, \dotsc} \cap \set {1, 2, 3, 4}$

$\ds $

$=$

$\ds \set {1, 2, 3, 4}$

$\ds \paren {A \cap C} \cup \paren {B \cap C}$

$=$

$\ds \paren {\set {2, 4, 6, 8, \dotsc} \cap \set {1, 2, 3, 4} } \cup \paren {\set {1, 3, 5, 7, \dotsc} \cap \set {1, 2, 3, 4} }$

$\ds $

$=$

$\ds \set {2, 4} \cup \set {1, 3}$

$\ds $

$=$

$\ds \set {1, 2, 3, 4}$

$\blacksquare$

1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $1$