Indexed Family/Examples
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Examples of Indexed Families
Arbitrary Set of Functions
Consider the set of functions $S = \set {f_1, f_2, f_3}$.
The indexing set for $S$ is $\set {1, 2, 3}$.
Arbitrary Sets of Students
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.