# Set Difference with Set Difference is Union of Set Difference with Intersection

## Theorem

Let $R, S, T$ be sets.

Then:

$R \setminus \paren {S \setminus T} = \paren {R \setminus S} \cup \paren {R \cap T}$

where:

$S \setminus T$ denotes set difference
$S \cup T$ denotes set union
$S \cap T$ denotes set intersection.

### Corollary

$T \setminus \paren {S \setminus T} = T$

## Proof

Consider $R, S, T \subseteq \mathbb U$, where $\mathbb U$ is considered as the universe.

 $\displaystyle R \setminus \paren {S \setminus T}$ $=$ $\displaystyle R \cap \overline {\paren {S \cap \overline T} }$ Set Difference as Intersection with Complement $\displaystyle$ $=$ $\displaystyle R \cap \paren {\overline S \cup \overline {\paren {\overline T} } }$ De Morgan's Laws $\displaystyle$ $=$ $\displaystyle R \cap \paren {\overline S \cup T}$ Complement of Complement $\displaystyle$ $=$ $\displaystyle \paren {R \cap \overline S} \cup \paren {R \cap T}$ Intersection Distributes over Union $\displaystyle$ $=$ $\displaystyle \paren {R \setminus S} \cup \paren {R \cap T}$ Set Difference as Intersection with Complement

$\blacksquare$