Use quantifiers to express:"There is someone who loves no one besides himself or herself"

Question

If $L(x,y)$: x loves y.

Use quantifiers to express:

"There is someone who loves no one besides himself or herself"

The answer given by textbook is $∃x∀y(L(x,y) ↔ x=y)$

But I think $∃x∀y(L(x,y) → x=y)$ is also correct, right?

Answer 1

It depends on how you interpret the English sentence "There is someone who loves no one besides himself or herself". Does the sentence imply that this someone loves himself/herself? If no, your answer is correct; if yes, the textbook's answer is correct.