# Intersection Distributes over Symmetric Difference

## Theorem

$\paren {R * S} \cap T = \paren {R \cap T} * \paren {S \cap T}$
$T \cap \paren {R * S} = \paren {T \cap R} * \paren {T \cap S}$

## Proof

From Set Intersection Distributes over Set Difference, we have:

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

So:

 $\displaystyle \paren {R \cap T} * \paren {S \cap T}$ $=$ $\displaystyle \paren {\paren {R \cap T} \setminus \paren {S \cap T} } \cup \paren {\paren {S \cap T} \setminus \paren {R \cap T} }$ Definition of Symmetric Difference $\displaystyle$ $=$ $\displaystyle \paren {\paren {R \setminus S} \cap T} \cup \paren {\paren {S \setminus R} \cap T}$ Set Intersection Distributes over Set Difference $\displaystyle$ $=$ $\displaystyle \paren {\paren {R \setminus S} \cup \paren {S \setminus R} } \cap T$ Intersection Distributes over Union $\displaystyle$ $=$ $\displaystyle \paren {R * S} \cap T$ Definition of Symmetric Difference

The second part of the proof is a direct consequence of the fact that Intersection is Commutative.

$\blacksquare$