Translate to first order logic: There is somebody who likes everyone who doesn’t like themself

Question

Translate to first order logic: There is somebody who likes everyone who doesn’t like themself

My answer is $∃x∀y( x ≠ y → love(x, y))$ but I'm not sure

Answer 1

I will define $love(x,y)$ to be true if $x$ loves $y$, and false if $x$ does not love $y$.

Then the statement would become:

$$\exists x \forall y\big(\neg love(y, y) \rightarrow love(x, y)\big)$$

This translates to: There exists $x$ such that for all $y$, if $y$ does not love $y$, then $x$ loves $y$. You can see that the "for all" in front of the $y$ is actually redundant because it is implicit as the domain for $y$.

Does that make sense? What did you mean by $x\neq y$?

Answer 2

The statement can be rewritten as: "There exists an $x$ such that for all $y$, if $y$ does not love itself, then $x$ loves $y$." In logical expression, $\exists x, \forall y, (\neg love(y,y) \implies love(x,y))$. This says that there is some person $x$ who loves all people $y$ who don't love themselves.