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
\(\ds \) | \(\) | \(\ds \paren {A \cap B} \cup \paren {C \cap A} \cup \relcomp {\Bbb U} {\relcomp {\Bbb U} A \cup \relcomp {\Bbb U} B}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {A \cap B} \cup \paren {C \cap A} \cup \relcomp {\Bbb U} {\relcomp {\Bbb U} {A \cap B} }\) | 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 |
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $12$