Predicate logic/Friend/Exercise

The expression ${}F(x,y)$ means that ${}x$ is a friend of ${}y$. We consider the sentence "All friends of Paula (${}P$) are also friends of Susanna (${}S$).“ Answer, for each of the following formulations, what they mean in usual language, whether they are true, and whether they express the mentioned sentence (the quantifiers are with respect to the set of all people).

1. $\forall x\exists yF(x,y),$

2. $\forall xF(P,x)\rightarrow \forall xF(S,x),$

3. $\forall x\forall y{\left(F(x,y)\rightarrow F(y,x)\right)},$

4. $\forall x\forall y{\left(F(x,y)\rightarrow F(x,y)\right)},$

5. $\exists x\forall yF(x,y),$

6. $\forall x{\left(F(P,x)\rightarrow F(x,S)\right)},$

7. $\forall x{\left(\forall y{\left(F(x,y)\rightarrow \forall zF(x,z)\right)}\right)},$

8. $\forall xF(x,P)\rightarrow \forall xF(x,S),$

9. $\forall x{\left(F(x,P)\rightarrow F(x,S)\right)},$

10. $\forall x{\left(\forall yF(x,y)\rightarrow F(x,x)\right)},$

11. $\forall xF(x,x),$

12. $\exists x\forall y{\left(\neg F(x,y)\right)}.$

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