Are my attempts to write "There exists a student studying all the subjects of the information technology subject" in FOL correct?

Question

I have the following sentence, which I need to write in FOL

There exists a student studying all the subjects of the information technology subject

I don't know how $\forall$ can be combined with the $\land$.

It was written by my teacher as follows

$$\exists x, \forall y: \text{student}(x) \land \text{learn}(x, y) \land \text{ITsubject}(y)$$

However, I was thinking this sentence should be written like this:

$$\exists x \text{student}(x) \land \forall y(\text{ITsubject}(y) \rightarrow \text{learn}(x, y))$$

or maybe

$$\exists x \text{student}(x) \land \forall y(\text{learn}(x, y) \rightarrow \text{ITsubject}(y))$$

Is this right?

Answer 1

Your two trials are obviously wrong since the occurrence of the latter $x$ are both free if you pay attention to its scope and paranthesis, and your last trial cannot be salvaged since such student may possibly learn non-IT subjects too. You can salvage your first trial as $∃x (student(x) \land ∀y(ITsubject(y) \to learn(x, y)))$, which could be further converted to prenex normal form $∃x∀y (student(x) \land (ITsubject(y) \to learn(x, y)))$ or $∃x∀y (student(x) \land (\lnot ITsubject(y) \lor learn(x, y)))$. And from this it shows your teacher's translation is incorrect or there's some typo in between, which is also apparent semantically since your teacher's version means every subject in your universe is ITsubject which is trivial here and most likely false in most 1-sorted context.