Intersection is Idempotent

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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 $S$:

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


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

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


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


Also see