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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.

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    $\begingroup$ This question seems more appropriate on the Math StackExchange: math.stackexchange.com $\endgroup$ – Derek Elkins Jun 1 '17 at 12:33
  • $\begingroup$ Could you try to reformat your question using MathJax? $\endgroup$ – Roukah Jun 1 '17 at 13:11
  • $\begingroup$ @Roukah The question seems pretty readable as it is. $\endgroup$ – David Richerby Jun 1 '17 at 14:56
  • $\begingroup$ @DerekElkins Logic is on-topic, here. $\endgroup$ – David Richerby Jun 1 '17 at 14:56
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    $\begingroup$ We discourage "please check whether my answer is correct" questions, as only "yes/no" answers are possible, which won't help you or future visitors. See here and here. Can you edit your post to ask about a specific conceptual issue you're uncertain about? As a rule of thumb, a good conceptual question should be useful even to someone who isn't looking at the problem you happen to be working on. If you just need someone to check your work, you might seek out a friend, classmate, or teacher. $\endgroup$ – David Richerby Jun 1 '17 at 14:57
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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:

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

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