First Order Logic : Predicates
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:

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