Need Help with Predicate Logic( True Or False)

Having not studied Logic formally, I have no understanding at all of the symbols or application of them that you show in a. and b. That’s a total mystery to me, and I’m not going to attempt to figure it out for this response.

But in your A. B. C. syllogisms:
B can be a subset of A. Perhaps all of the boys like Mary best.

A and C are mutually exclusive, obviously.

B and C are not necessarily exclusive, since B says nothing about what “else” boys might have done. Aside from kissing the girl they liked best, they may have kissed all of the other girls, too, and even had a go at each other. The statement is silent on any ‘else’ behavior. So every boy can kiss the girl he likes best (either Mary or Sue), and then also kissed the other girl.

So B can also be a subset of C.

How are x and y assigned? Is x – kisses and y – likes?

OP: The wording of your question is unclear *, but the logical notation seems straightforward to me.

(a) says, “Every boy kisses some girl.” (For all x, if x is a Boy, then there is some y such that y is a Girl and x Kisses y.)
(b) says, “There is some girl that every boy kisses.” (There is some y such that y is a Girl and for all x, if x is a Boy, then x Kisses y.)

So now look at A, B, and C, and compare it to (a) and (b). I’m assuming that the question concerns whether A, B, and C are consistent with (a) and (b), but I’m not sure because your wording is unclear. So I’ll leave the rest to you.

* “In which situation a is definitely true, and do the same for b” is ungrammatical.

a. ∀x(Bx →∃y(Gy ^ Kxy))
For each x, if x is a boy, then there is a girl y such that x kissed y.
Every boy has kissed at least one girl.

b. ∃y(Gy ^∀x(Bx → Kxy))
There is a y such that y is a girl and for each x, if x is a boy, then x has kissed y.
At least one girl has been kissed by every boy.

Edit:

I’m assuming that the question concerns whether A, B, or C are consistent with, imply, or are implied by (a) or (b).

I’m not sure I get the question but…

(a) says “every boy kissed at least one girl.” It doesn’t mean they kissed the same girl, but they might have.
A: every boy kissed Mary, so (a) is true.
B: every boy kissed a girl, the one they like the most. so (a) is true. [unless there are ties. e.g. if a boy has two favorite girls, then there may not be a girl he likes the most.]
C: every boy kissed Mary AND Sue. so (a) is true.

(b) says “there is at least one girl who has kissed all the boys”
A: Mary kissed them all. so (b) is true.
B: if the boys like different girls the most, then maybe no girl has kissed all the boys. so (b) is false.
C: Mary and Sue kissed all the boys. so (b) is true.

Does this help? Are you able to translate the symbolic statements into English? If you can do that, it helps with stuff like this a lot.