Indexed Family/Examples/Arbitrary Sets of Students

Example of Indexed Family
Let $S$ be the set of students at a given university.

Let:
 * $A_1$ denote the set of first year students
 * $A_2$ denote the set of second year students
 * $A_3$ denote the set of third year students
 * $A_4$ denote the set of fourth year students.

We have:
 * $I = \set {1, 2, 3, 4}$ is an indexing set.

Hence $\alpha: I \to S$ is an indexing function on $S$.

Hence:
 * $\ds \bigcup_{\alpha \mathop \in I} A_\alpha = $ the set of all undergraduates at the university

and:
 * $\ds \bigcap_{\alpha \mathop \in I} A_\alpha = \O$

where $\ds \bigcup_{\alpha \mathop \in I} A_\alpha$ and $\ds \bigcap_{\alpha \mathop \in I} A_\alpha$ denote the union of $\family {A_\alpha}$ and intersection of $\family {A_\alpha}$ respectively.