Intersection of a family of sets

Definition

Let $(A_i)_{i \in I}$ be a family of sets with a nonempty index set $I$. The intersection of the family is the set $$\bigcap_{i \in I} A_i = \{x \mid \forall i \in I, \, x \in A_i\}.$$ So $x$ is an element of the intersection when $x \in A_i$ for each $i \in I$.

Remarks

For $I = \{1, 2\}$, this is the intersection $A_1 \cap A_2$. For an empty index set, the condition is vacuously true for each $x$. So the intersection is defined only relative to a universal set $U$, and it is $U$. Example: $\bigcap_{n \ge 1} (-1/n, 1/n) = \{0\}$ in $\mathbb{R}$.

Explanations

L1 human

The intersection of a family keeps only the elements that are in every member of the family. It generalizes $A \cap B$ from two sets to any number of sets.