"Everyone who Patricia likes, Sue doesn't like"
Let $L(x,y)$ stand for "$x$ likes $y$" and $p,s$ for Patricia and Sue, respectively.
Then the statement in logic is:
$\forall x (L(p,x) \implies \neg L(s,x))$
$= \forall x(\neg L((p,x) \vee \neg L(s,x) )$
$=\forall x \neg(L(p,x)\wedge L(s,x))$
$= \neg \exists(L(p,x) \wedge L(s,x))$
I don't understand the first statement of equivalence, $\forall x(\neg L((p,x) \vee \neg L(s,x) )$, and I also don't understand why the original statement in logic is valid, since:
$\neg P \vee \neg Q = \neg(P \wedge Q)$, and a true hypothesis cannot imply a false conclusion: $P \implies \neg Q$ is invalid, right? since the conclusion must be true if the hypothesis (premise) is true, in other words, the conclusion is "forced" to be true by the premise, right?
So if $P \implies \neg Q$ is invalid, then how can the original statement, $\forall x (L(p,x) \implies \neg L(s,x))$, which is the form of an invalid argument, be valid? $P$ is true, yet $Q$ is false. And how is the original statement in logic equivalent to the first equivalence, $\forall x(\neg L((p,x) \vee \neg L(s,x) )$?
The original sentence is: "Everyone who Patricia likes, Sue does not like."
The first statement in logic reads: "For every person $X$ in the set the following is true: if $X$ is liked by Patricia, then $X$ is not liked by Sue."
The next statement in logic reads: "For every person $X$ in the set the following is true: $X$ is not liked by Patricia, or not liked by Sue, or not liked by both Patricia and Sue."
Clearly all three statements are equivalent.