Definition:Set Intersection/Family of Sets/Two Sets

Definition
Let $I = \set {\alpha, \beta}$ be an indexing set containing exactly two elements.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.

From the definition of the intersection of $S_i$:
 * $\ds \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$

it follows that:
 * $\ds \bigcap \set {S_\alpha, S_\beta} := S_\alpha \cap S_\beta$