# Mapping/Examples/Arbitrary Sets of Students

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## Example of Mapping

Consider the indexed family as defined in 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.

The indexing function $\alpha: I \to S$ which establishes a correspondence between elements of $I$ and $S$ is an example of a mapping.

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 6$: Functions