# Symmetric Difference is Subset of Union of Symmetric Differences

## Theorem

Let $R, S, T$ be sets.

Then:

$R * S \subseteq \left({R * T}\right) \cup \left({S * T}\right)$

where $R * S$ denotes the symmetric difference between $R$ and $S$.

## Proof

From the definition of symmetric difference, we have:

$R * S = \left({R \setminus S}\right) \cup \left({S \setminus R}\right)$

Then from Set Difference is Subset of Union of Differences, we have:

$R \setminus S \subseteq \left({R \setminus T}\right) \cup \left({T \setminus S}\right)$
$S \setminus R \subseteq \left({S \setminus T}\right) \cup \left({T \setminus R}\right)$

Thus:

 $\displaystyle \left({R \setminus S}\right) \cup \left({S \setminus R}\right)$ $\subseteq$ $\displaystyle \left({R \setminus T}\right) \cup \left({T \setminus S}\right) \cup \left({S \setminus T}\right) \cup \left({T \setminus R}\right)$ Set Union Preserves Subsets $\displaystyle$ $=$ $\displaystyle \left({R \setminus T}\right) \cup \left({T \setminus R}\right) \cup \left({S \setminus T}\right) \cup \left({T \setminus S}\right)$ Union is Commutative $\displaystyle$ $=$ $\displaystyle \left({R * T}\right) \cup \left({S * T}\right)$ Definition of Symmetric Difference

$\blacksquare$