Intersection with Set Difference is Set Difference with Intersection/Proof 1

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Theorem

$\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus S$


Proof

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


Then:

\(\displaystyle \left({R \setminus S}\right) \cap T\) \(=\) \(\displaystyle \left({R \cap \complement \left({S}\right)}\right) \cap T\) Set Difference as Intersection with Complement
\(\displaystyle \) \(=\) \(\displaystyle \left({R \cap T}\right) \cap \complement \left({S}\right)\) Intersection is Commutative and Intersection is Associative
\(\displaystyle \) \(=\) \(\displaystyle \left({R \cap T}\right) \setminus S\) Set Difference as Intersection with Complement

$\blacksquare$