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:
\(\ds \paren {R \setminus S} \cap T\) | \(=\) | \(\ds \paren {R \cap \map \complement S} \cap T\) | Set Difference as Intersection with Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {R \cap T} \cap \map \complement S\) | Intersection is Commutative and Intersection is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {R \cap T} \setminus S\) | Set Difference as Intersection with Complement |
$\blacksquare$