Converting a statement This is from "How to Prove It: A Structured Approach" by Daniel J. Velleman. In the selected exercise the goal was to negate the statement, but I'm more interested in its form.
The statement is: Everyone has a roommate who dislikes everyone. 
Let $R(x,y)$ stand for $y$ is $x$'s roomate and $L(x,y)$ stand for $x$ likes $y$. Then the statement becomes:
$\forall x \exists y(R(x,y) \land \forall z \neg L(y,z))$.
My question is - why is this the case?
Why wouldn't a conditional statement be used here such that that the statement becomes
$\forall x \exists y(R(x,y) \rightarrow \forall z \neg L(y,z))$.
What would this statement mean if it were translated into English?
 A: In the first case, everyone has a roommate, not in the second case. If we translate your proposal:
For every person $x$, there exists and another person $y$ such that if this person is the rommate of $x$, he dislikes everyone.
We could say that everyone has a bad influence on at least one person...
A: "Every person X has a person named Y who, if X and Y are roommates, then Y dislikes everyone." Notice that this is always true as long as there is someone who isn't X's roommate: if I were X, for instance, then I could take Y to be Hilary Clinton and the statement would hold (for I am not roommates with Hilary Clinton).
A: In the original sentence, the roommate who dislikes everyone does so regardless of who his own roommates are.  So $L(y,z)$ is independent of $R(x,y)$.
Remember that 
$$\phi \to \psi \equiv \neg (\phi \wedge \neg \psi) \equiv \neg\phi \vee \psi$$
So
$$
\forall x\exists y(R(x,y)\to\forall z \neg L(y,z))
\equiv
\forall x\exists y(\neg R(x,y) \vee \exists z L(y,z))
$$
This translates to: For every person x, there exists a person y who is not x's roommate or likes a person z.  Like Patrick says, this is satisfied as long as there are two people who are not roommates.
