Complement of Relative Complement is Union with Complement

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Theorem

Let $A, B, C$ be sets such that $A \subseteq B \subseteq C$.

Then:

$\relcomp C {\relcomp B A} = A \cup \relcomp C B$


Proof

\(\displaystyle \relcomp C {\relcomp B A}\) \(=\) \(\displaystyle C \setminus \paren {B \setminus A}\) Definition of Relative Complement
\(\displaystyle \) \(=\) \(\displaystyle \paren {C \setminus B} \cup \paren {C \cap A}\) Set Difference with Set Difference is Union of Set Difference with Intersection
\(\displaystyle \) \(=\) \(\displaystyle \paren {C \setminus B} \cup A\) Intersection with Subset is Subset
\(\displaystyle \) \(=\) \(\displaystyle A \cup \paren {C \setminus B}\) Union is Commutative
\(\displaystyle \) \(=\) \(\displaystyle A \cup \relcomp C B\) Definition of Relative Complement

$\blacksquare$


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