Intersection is Idempotent

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Theorem

Set intersection is idempotent:

$S \cap S = S$


Indexed Family

Let $\family {F_i}_{i \mathop \in I}$ be a non-empty indexed family of sets.

Suppose that all the sets in the $\family {F_i}_{i \mathop \in I}$ are the same.

That is, suppose that for some set $S$:

$\forall i \in I: F_i = S$


Then:

$\ds \bigcap_{i \mathop \in I} F_i = S$

where $\ds \bigcap_{i \mathop \in I} F_i$ is the intersection of $\family {F_i}_{i \mathop \in I}$.


Proof

\(\ds x\) \(\in\) \(\ds S \cap S\)
\(\ds \leadstoandfrom \ \ \) \(\ds x \in S\) \(\land\) \(\ds x \in S\) Definition of Set Intersection
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds S\) Rule of Idempotence: Conjunction

$\blacksquare$


Also see


Sources