I'm working through some exercises in preparation of a test, I just have the answers, no motivation for why something is wrong. Which is essential in learning imo, so I hope you could give some pointers

Given:

J: John.

P: Pete.

C(x, y): x is a child of y.

M(x): x is a man.

W(x): x is a woman.

Write as a predicate:

John or Pete has a child:

My solution: $\displaystyle \exists x(C(x, J) \lor C(x, P)) $.

The solution given is:$\displaystyle \exists x(C(x, J)) \lor \exists x(C(x, P)$.

Is my solution wrong? And so, why? Am I not allowed to use the scope of the quantifier like this in this context? Is my solution solution actually read as 'There's a child that belongs to either John or Pete' by using scope like this?

Everybody has a father and mother:

My solution: $\displaystyle \forall x(\exists x(W(z) \land C(x, z)) \land \exists y(M(y) \land C(x, y))$

The given solution: $\displaystyle \forall x \exists y,z(W(z) \land C(x, z) \land M(y) \land C(x, y))$

Again, basically the same question: is my use of scoping correct? Should I prefer to rewrite with the quantifiers out? Is this like not simplifying a fraction?