How to define a set as a subset with the property relative to set i.e. standard deviation

Question

So i'm reading a book on applied category theory and while i can understand how to define a subset with a simple property i.e. "all even numbers" i'm curious how you define it based in a more real work scenario. As in if you had the set of people (maybe a "class of students") how would you define a subset that is a set of all people who's age is 1 standard deviation from the mean? or am i thinking about thing wrong?

Answer 1

Let's take your specific example. We have a set of students, and we would like to define the subset consisting of all people who's age is within 1 standard deviation from the mean. Here is how this set might be described symbolically:

Let $S$ denote a set of students. For a student $s \in S$, let $\operatorname{age}(s)$ denote the age of that student. We define the set of ages by $$ A = \{a \in \Bbb N : \text{there is a student } s \in S \text{ such that} \operatorname{age}(s) = a\}. $$ Let $\mu,\sigma$ denote the mean and standard deviation of $A$, respectively. We then define $$ S = \{s \in S: |\operatorname{age}(s) - \mu| \leq \sigma\}. $$

In this context, $S$ would be the set that we were trying to define.