Are these statements in logic form correct?

Question

Let M represent the set of all Mathematics courses and S represent the set of all students.

Predicates:

E(s, m) : "student s is enrolled in math course m"

C(s): "student s owns a cat"

I want to transform the two sentences below using quantified predicates.

1. There is at least one math student who owns a cat. (A math student is a student who is registered in at least one math class)

2. There is at least one student in every math class who owns a cat.

I got this:

1) ∃s∈M, E(s, m)-> C(s)

or perhaps

∃E(s,m), C(s)

Is this correct?

2) ∃s∈M, ∀m∈M, C(s) ?

Also for:

∀s∈S, ∀m∈M, [E(s,m) -> ¬C(s)]

does this translate to:

"All student cat owners are not registered in any math courses"

Thank you.

Answer 1

For the first question, use something like $$\exists s \exists m(E(s,m)\land C(s)).$$ If you want to specify membership, that's fine, but one should be consistent. So you could write $$(\exists s\in S)( \exists m\in M)(E(s,m)\land C(s)).$$

For the second, $$\forall m\exists s(E(s,m)\land C(s))$$ will work.

Your version of (2) is not correct. Since you put the existential quantifier in front, it says it says in particular that there is a particular student who is enrolled in every single math class. Very busy.

On the third question, the translation is correct. Perhaps it could be made a little more idiomatic. Something like "No math student owns a cat."

Answer 2

For (1), the formula "$\exists s \in M$, $E(s, m)\to C(s)$" translates to "there is a math course $s$ such that if $s$ is enrolled in $m$, then $s$ has a cat". This is not what you wanted to say. And the formula $\exists E(s,m), C(s)$ makes no sense because quantifiers can only be applied to variable symbols, not predicate symbols.

To write "there is at least one math student who owns a cat," where a math student is a student who is enrolled in at least one math class, we need to write "there is at least one student who is enrolled in at least one math class and who owns a cat." This can be done as $\exists s\,\exists m, (E(s,m) \wedge C(s))$ as André says, or alternatively as $\exists s, ((\exists m, E(s,m)) \wedge C(s))$ because "$\exists m, E(s,m)$" says that $s$ is a math student.

For (2), the formula "$\exists s\in M, \forall m \in M, C(s)$" says "there is a math course $s$ such that for all math courses $m$, $s$ owns a cat. Again not what you wanted to say: for one thing, courses cannot own cats.

For your third question ("Also for:") the statement "$\forall s \in S, \forall m \in M, (E(s,m) \to \neg C(s))"$ translates to "for all students $s$ and math courses $m$, if $s$ is enrolled in $m$, then $s$ does not have a cat." This is equivalent to "no math student has a cat," which in turn is equivalent to what you wrote.