Set Difference is Right Distributive over Set Intersection/Proof 2
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Theorem
- $\paren {A \cap B} \setminus C = \paren {A \setminus C} \cap \paren {B \setminus C}$
Proof
\(\ds x\) | \(\in\) | \(\ds \paren {A \cap B} \setminus C\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds A \land x \in B\) | Definition of Set Intersection | ||||||||||
\(\, \ds \land \, \) | \(\ds x\) | \(\notin\) | \(\ds C\) | Definition of Set Difference | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds A \land x \in B\) | Rule of Idempotence | ||||||||||
\(\, \ds \land \, \) | \(\ds x\) | \(\notin\) | \(\ds C \land x \notin C\) | Rule of Idempotence | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds A\) | Rule of Association | ||||||||||
\(\, \ds \land \, \) | \(\ds x\) | \(\in\) | \(\ds x \in B \land x \notin C\) | |||||||||||
\(\, \ds \land \, \) | \(\ds x\) | \(\notin\) | \(\ds C\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds A\) | |||||||||||
\(\, \ds \land \, \) | \(\ds x\) | \(\notin\) | \(\ds C \land x \in B\) | Rule of Commutation | ||||||||||
\(\, \ds \land \, \) | \(\ds x\) | \(\notin\) | \(\ds C\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds A \land x \notin C\) | Rule of Association | ||||||||||
\(\, \ds \land \, \) | \(\ds x\) | \(\in\) | \(\ds B \land x \notin C\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds \paren {A \setminus C}\) | Definition of Set Difference | ||||||||||
\(\, \ds \land \, \) | \(\ds x\) | \(\in\) | \(\ds \paren {B \setminus C}\) | Definition of Set Difference | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds \paren {A \setminus C} \cap \paren {B \setminus C}\) | Definition of Set Intersection |
$\blacksquare$