Intersection with Set Difference is Set Difference with Intersection/Proof 2
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Theorem
- $\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus S$
Proof
\(\ds \) | \(\) | \(\ds x \in \paren {R \setminus S} \cap T\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(\) | \(\ds \paren {x \in R \land x \notin S} \land x \in T\) | Definition of Set Intersection and Definition of Set Difference | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(\) | \(\ds \paren {x \in R \land x \in T} \land x \notin S\) | Rule of Commutation and Rule of Association | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(\) | \(\ds x \in \paren {R \cap T} \setminus S\) | Definition of Set Intersection and Definition of Set Difference |
$\blacksquare$