I am going back and reviewing some elementary material in logic/set theory. Among the problems in the quantifier section is the following english sentence to logic sentence translation:
Everyone likes Mary, except Mary herself.
Now, my attempt was: (∀x)[(x≠m → L(x,m)) ∧ (x=m → ¬L(x,m))], where m stands for "mary," and L(a,b) stands for "a likes b."
The solution to the exercise in the text was given as: (∀x)(x≠m → L(x,m)). I was a bit skeptical of this and searched the problem online and found another, different solution:
(∀x)(x≠m ↔ L(x,m)).
My question is, which of these three is correct? (or, if they are all/some equivalent, how?)