“Some students in this class grew up in the same town as exactly one other student in this class."
I'm thinking there is a relation T(x,y) where the student x grew up the same town as student y. And "some student" would imply ∃x. But my conclusion of ∃x(T(x,y)) seems too easy. Would there be a variable for the town? For specifying "exactly one student"? Thanks ahead.
You're on the right track, but we need to tighten our conditions.
Let $T(x,y)$ represent "$x$ grew up in the same town as $y$."
We're discussing two students, so we have $\exists x \exists y(T(x,y))$. However, we need to make sure person $x$ is not the same person as $y$, since every person grows up in the same town as themselves. So, we have $\exists x \exists y (T(x,y) \land x\not=y)$. Now we want for there to be exactly one $y$. This involves the uniqueness quantifier, so we can just write $\exists x \exists! y (T(x,y) \land x\not=y)$ and be done with it. However, we can rewrite this using the reduction formulas in the link. So, one possible answer is: $$\exists x \exists y [T(x,y) \land (x\not=y) \land \forall z(T(x,z)\rightarrow (z=y)].$$
This says that there are distinct persons $x$ and $y$ such that $x$ lives in the same town as $y$; AND if $x$ happens to live in the same town as any person $z$, then $z$ in fact, better be the same person as $y$.