"There are exactly two people that Eun likes"
Let $Q(x,y):$ $x$ likes $y$
My solution to this is:
$$\forall x (\lnot Q(Eun,x)\land\exists y,z((x\neq y\neq z) \rightarrow Q(Eun,y)\land Q(Eun,z)))$$
My thought was if Eun likes exactly two people, it means that he doesn't like everybody else. My problem with this is that you usually don't see an existential quantifier with a conditional...
Any thoughts?
I would try something like this: $$\exists x\exists y(\lnot(x=y)\land Q(E,x)\land Q(E,y)\land \forall t(Q(E,t)\longrightarrow ((t=x)\lor (t=y)))).$$
Some parentheses have been left out, since though technically needed, they would interfere with readability.
The first part says there are two different people that $E$ likes. The part that begins with $\forall t$ says $E$ likes nobody else.