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

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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

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

$\blacksquare$


Sources