All girls are liked by some boys. No boy likes a sulk. No girl is a sulk. 
*

*All girls are liked by some boys. 

*No boy likes a sulk. 

*No girl is a sulk.


How do you write the above in predicate logic statements without starting with a negation? My guess is below. Let me know if you differ. 


*

*∀x ∃y girl(x) → boy(y) ∧ like(y,x)  

*∀x ∃y boy(x) → sulk(y) ∧ ~like(x,y)  

*∀x girl(x) → ~ sulk(y)


After that, how do you prove it?
 A: 
How do you write the above in predicate logic statements without starting with a negation? 

First write them starting with a negation, then apply DeMorgan's Laws to "move it inwards".  Also use Implication Equivalence when applicable.
$${\text{No boy likes a sulk}\\\neg\exists x~\exists y~(B(x)\wedge S(y)\wedge L(x,y))\\\forall x~\neg\exists y~(B(x)\wedge S(y)\wedge L(x,y))\\\forall x~\forall y~\neg(B(x)\wedge S(y)\wedge L(x,y))\\\forall x~\forall y~(\neg(B(x)\wedge S(y))\vee\neg L(x,y))\\\forall x~\forall y~(B(x)\wedge S(y)\to\neg L(x,y))}$$
$\forall x~\forall y~(B(x)\to(S(y)\to\neg L(x,y)))$ is also acceptable. (Re: the exportation identity)
A: I would do it the following way: 
$1. \forall x \exists y  \exists z(G(x) \to ((y \neq z) \land L(yx) \land L(zx))
$, because it seems that there's more than one boy per girl satisfying that the former likes the latter.
$2. \forall x \forall y ((B(x) \land S(y)) \to \neg L(xy))$, because given a boy and a ulk, the boy wouldn't like the sulk (it doesn't matter if the sulk is a girl, because I'm progressive politically).
$3. \forall x (G(x) \to \neg S(x))$
Here $G$, $B$ and $S$ are the girl, boy and sulk predicates, respectively, and $L$ is the relation meaning that the first likes the second.
A: I think 2. and 3. should be of the form 


*$\forall x \forall y \;\text{boy}(x)\land\text{sulk} (y) \rightarrow \lnot \;\text{like}(x,y)$
and 


*$\forall x\; \text{girl}(x) \rightarrow \lnot\;\text{sulk}(x)$
A: What you have put here is almost all correct. Make sure you read up on the rules of logic so that you know what is going on here.


*

*All girls are liked by some boys. This is equivalent to:


$$\forall x ,G_x \implies \exists y \ni(B_y \wedge L_{(x,y)})$$


*No boy likes a sulk. This is equivalent to:


$$\neg \exists x,y\ni B_x \wedge S_y\wedge L_{(x,y)}$$
$$\forall x,y,\neg( B_x \wedge S_y\wedge L_{(x,y)})$$
$$\forall x,y,\neg B_x \vee \neg S_y \vee \neg L_{(x,y)})$$
$$\forall x,y, B_x \implies (\neg S_y \vee \neg L_{(x,y)})$$
$$\forall x,y, B_x \implies (S_y \implies \neg L_{(x,y)})$$
$$\forall x,y, (B_x \wedge S_y) \implies \neg L_{(x,y)}$$


*No girl is a sulk. This is equivalent to:


$$\neg \exists x \ni G_x \wedge S_x$$
$$\forall x, \neg (G_x \wedge S_x)$$
$$\forall x, \neg G_x \vee \neg S_x$$
$$\forall x, G_x \implies \neg S_x$$
