I have the following predicate exersice:
Explain why the two following sentences have different meanings (by a description of a situation in which the sentences have a different truth-value)
There is exactly one boy who loves exactly one girl. $(\exists!x \exists!y (Bx\land Gy\land Lxy))$
There is exactly one girl who is loved by exactly one boy. $(\exists!y \exists!x (Bx \land Gy \land Lxy))$
The answer is given as follows:
I see here two guys and girls (being black or white dots) and a guy having a love relation to a girl, and a guy having a love relation to two different girls.
I don't see how this fits either formula. Or even how the switching of $\exists!x \exists!y$ to $\exists!y \exists!x$ has any impact on the meaning of the predicate.
Consider the following four scenarios:
Here there is a unique boy who loves a single girl. There is not a unique girl who is loved only once.
Here there is not a unique boy who loves only one girl. There is a unique girl who is loved only once.
Here there is both a unique boy who loves a single girl and a single girl who is loved only once.
Here there are no boys who love only one girl and there are no girls who are loved only once.
As for the image provided as the "answer", it certainly isn't a clear answer but I interpret it as trying to get across the idea that a unique boy loving one girl is a result of the far left black dot and it's interaction with the first white dot. A unique girl being loved by a unique boy is a result of the far right white dot being loved by the right black dot. Had these expressions been the same, then surely it would have been a result of the same dots, not different ones.
In the image, both expressions have the same truth value. If their goal was to show a situation where the expressions have different truth values, then they failed and their "answer" is incorrect.