Some students in this class grew up in the same town as at least two other students in this class.
I'm thinking that I'm required to use the following information:
$T(x,y):$ student $x$ grew up in town $y$
$C(x):$ student $x$ is in this class.
I think "some students" means $∃x C(x).$ But my translation $$∃x C(x) T(x,y)$$ seems too naive. How is it possible to express "at least two other students" using quantifiers? What is a more accurate translation?
Let $D=\{ x: x$ is a person$\}$ be the domain consisting of all people. We also define the following predicate symbols...
$ \begin{array}{11} Txy: & \text{$x$ grew up in the same town as $y$} \\ Cx: & \text{$x$ is in this class} \\ \end{array} $
To translate your statement properly, there are at least three distinct students that need to be specified:
"Some students in this class [that is, at least one student in this class, whom we call $x$] grew up in the same town as at least two other students in this class [whom we call $y$ and $z$]."
It is necessary to specify $y \neq z$ because there are at least two students, and it is necessary to specify $x \neq y$ and $x \neq z$ because there are at least two other students.
$$ \exists x \exists y \exists z[Cx \wedge Cy \wedge Cz \wedge x \neq y \wedge x \neq z \wedge y \neq z \wedge Txy \wedge Txz] $$