Symmetric Difference is Subset of Union of Symmetric Differences

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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

$\blacksquare$