Nested Quantifiers, "Unique" or "Exactly One" Example

Question

Question

Let L(x, y) be the predicate: "x likes y", where the domains are given by: x is a CS student and y is a kind of food.

Let D(x) be the predicate: "x is a student in this discrete class", where the domain is: all CS students.

Express the following statement using those predicates and any required quantifiers. You may use only universal and existential quantifiers.

(a) There is exactly one CS student who likes tofu.

Response

My answer: ∃x∀y(L(x, tofu) ∧ ((y ≠ x) ⟹ ¬L(x, y))

What is wrong with my answer?

Answer 1

Let $t$ stand for tofu. Let $CS$ be the set of all CS majors.

Then \begin{align} & & &\exists! x \in CS \, L(x,t) & &\text{Exactly one CS student likes $t$.} \\[1ex] &\text{i.e.,} &\quad &\exists x \in CS \, \bigl[ L(x,t) \wedge \forall y \in CS \, (L(y,t) \to y = x ) \bigr] & &\begin{array} $\text{A CS student $x$ likes $t$, and every} \\ \text{CS student that likes $t$ must be $x$.}\end{array} \end{align}

Answer 2

There is at least one student $x$ that likes tofu such as all students in the CS class that are not $x$ don't like tofu. It's easy to show that $x$ is unique.