Set Difference is Right Distributive over Set Intersection/General Case

Theorem

Let $U$ be a collection of sets.

Let $T$ be a set.

Then:

$\ds \bigcap_{X \mathop \in U} \paren {X \setminus T} = \paren {\bigcap_{X \mathop \in U} X} \setminus T$

That is, the difference with an intersection equals the intersection of the differences.

Proof

\(\ds x\)

\(\in\)

\(\ds \bigcap_{X \mathop \in U} \paren {X \setminus T}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall X \in U: \, \)

\(\ds x\)

\(\in\)

\(\ds X \setminus T\)

Definition of Set Intersection

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall X \in U: \, \)

\(\ds x\)

\(\in\)

\(\ds X\)

Definition of Set Difference

\(\, \ds \land \, \)

\(\ds x\)

\(\not \in\)

\(\ds T\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds \bigcap_{X \mathop \in U}\)

Definition of Set Intersection

\(\, \ds \land \, \)

\(\ds x\)

\(\not \in\)

\(\ds T\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds \paren {\bigcap_{X \mathop \in U} X} \setminus T\)

Definition of Set Difference

$\blacksquare$