Intersection is Idempotent

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Theorem

Set intersection is idempotent:

$S \cap S = S$


Indexed Family

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

Suppose that all the sets in the family are the same.

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

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


Then:

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

where $\displaystyle \bigcap_{i \mathop \in I} F_i$ is the intersection of $\left\langle{ F_i }\right\rangle_{i \in I}$.


Proof

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

$\blacksquare$


Also see


Sources