# Set Theory/Examples/(A cap B) cup (C cap A) cup Complement (Complement A cup Complement B)

## Example in Set Theory

The expression:

$\paren {A \cap B} \cup \paren {C \cap A} \cup \relcomp {\Bbb U} {\relcomp {\Bbb U} A \cup \relcomp {\Bbb U} B}$

can be simplified to:

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

## Proof

 $\ds$  $\ds \paren {A \cap B} \cup \paren {C \cap A} \cup \relcomp {\Bbb U} {\relcomp {\Bbb U} A \cup \relcomp {\Bbb U} B}$ $\ds$ $=$ $\ds \paren {A \cap B} \cup \paren {C \cap A} \cup \relcomp {\Bbb U} {\relcomp {\Bbb U} {A \cap B} }$ De Morgan's Laws: Complement of Intersection $\ds$ $=$ $\ds \paren {A \cap B} \cup \paren {C \cap A} \cup \paren {A \cap B}$ Complement of Complement $\ds$ $=$ $\ds \paren {\paren {A \cap B} \cup \paren {A \cap B} } \cup \paren {A \cap C}$ Union is Commutative, Union is Associative, Intersection is Commutative $\ds$ $=$ $\ds \paren {A \cap B} \cup \paren {A \cap C}$ Set Union is Idempotent $\ds$ $=$ $\ds A \cap \paren {B \cup C}$ Intersection Distributes over Union

$\blacksquare$