# Complement of Relative Complement is Union with Complement

## 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

$\blacksquare$