First Order Logic : Predicates

Question

I have a small problem with the first order logic, in particular, predicate logic

Let us take this sentence as an example:

Each teacher has given a form to each student.

From this sentence, can we have different reading?

This is my try to solve such problem, I did not know if this is the answer for such question:

Every Teacher has given a form to each Student.

(∀x)Teacher(x)^(∀y)  Student(y)^(∃z)Form(z)^Give(x,y,z)

If X is a Student then he has received a form from a teacher

Student(x)→(∃y)  Teacher(y)^(∃z)Form(z)^Give(x,y,z)

If X is a Teacher then he has gave a from for all his students

Teacher(x)→(∀y)  Student(y)^(∃z)Form(z)^Give(x,y,z)

If X is a form then a teacher gave it to all student.

Form(x)→(∀y)  Employer(y)^(∃z)Teacher(z)^Give(x,y,z)

EDIT

To clarify my needs !

I think that the form is in logical dependency, either of student, either of teacher, or both, so we can find three other readings; so What are those reading? (Look at the begining to find what I did)

And How can I transform this sentence in predicate logic?

Each teacher has given a form to each student.

because it seems that

(∀x)Teacher(x)^(∀y)  Student(y)^(∃z)Form(z)^Give(x,y,z)

is wrong.

Answer 1

Each teacher has given a form to each student.

becomes

$$\forall x\forall y[\mathrm{Teacher}(x)\wedge\mathrm{Student}(y)\to\exists z(\mathrm{Form}(z)\wedge\mathrm{Give}(x,y,z))]$$ The main thing is to understand this table:

$\begin{eqnarray*}\text{every } A\quad & \forall x(A(x)\to\dots) \\ \text{some } A\quad & \exists x(A(x)\wedge\dots) \end{eqnarray*}$