How to write this set?

Question

I hope someone can help me out here :)

We have to sets : STUDENTS » All the students of the school CLASSES » All the classes of the school And the relation : STUDENTSCLASSES » Relates the students to the classes they attend to

We need to write the set that includes only the students who attend to the "Maths class".

Would it be something like this? : $\{student STUDENTSCLASSES class | class = Maths\}$

Or rather : $\{(student,class) \in STUDENTS X CLASSES | class = Maths\}$

Or what is not ?

I think my answers are wrong since the answer we are looking for is a set of elements, where the elements are just students..

Thanks !

Answer 1

Indeed, we have the set $S$: the set of all students, and $C$: the set of all classes. We want to denote the relation $A \subset S\times C$ as the set of ordered pairs $(s, m) \equiv s\;\text{ STUDENTCLASSES }\;m$, meaning student $s\in S$ is in a maths class $m\in C$. $$A = \{(s, m)\mid s\in S, m\in C, s\text{ STUDENTCLASSES } m\}$$

Now, if you want to denote just the set of students taking math classes, then indeed, you want $$\{s\mid (s \text{ STUDENTCLASSES } m) \land (s\in S) \land (m \in C)\} \\ \\= \{s\mid (s, m)\in A\}$$

Answer 2

If the student set is $S$, class set is $C$, and the relation "Student $s$ attends class $c$" as $R$, where$R \subseteq S \times C$: $$\{\text{all students who attend math class}\}=\{s\mid sRmath\}=\{s\mid(s,math) \in R\}$$

Answer 3

$STUDENTS$ is the set of students, so that $x \in STUDENTS$ means "$x$ is a student".

$CLASSES$ is the set of classes, so that $y \in CLASSES$ means "$y$ is a class".

$STUDENTCLASSES$ is the relation between students and classes, so that $x STUDENTCLASSES y$ means "the student $x$ attend the class $y$".

We have that :

$STUDENTCLASSES = \{ <s,y> : s \in STUDENT \land y \in CLASSES \}$

where $\land$ is the logical connective "and".

Math class is an element of the set $CLASSES$: call it $m$. We have that $m \in CLASSES$.

The set of students that attend Math class is

$X = \{ s : <s,m> \in STUDENTCLASSES \}$

NOTE I assume that you are familiar with the concept of ordered pair (in symbol : $<x,y>$).