Intersection Of A Collection Of Sets

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Definition (Intersection Of A Collection Of Sets)

If $A$ and $B$ are a two sets, we say that $x\in A\cap B$ if and only if $x\in A$ and $x\in B$. If $\mathscr{B}$ is a collection of sets, then we say that $x$ is an element of the interesection of the collection of sets if and only if $x\in B$ for each $B\in \mathscr{B}$. We write this in set builder notation as $$\ds\bigcap_{B\in \mathscr{B}}B = \{x\mid x\in B \text{ for each }B\in \mathscr{B} \}.$$

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