All students in a class are friends with some students in all engineering faculty 
All students in a class are friends with some students in all engineering faculty. 

My solution was Domain $x,y$ is a student Domain $z$ is engineering faculty $M(x)$ students in this class $S(x,y)$; $x$ is friends with $y$; $F(y,z)$ denotes the sentence "$y$ is from $z$." 
$$\forall x \ \exists y \ \forall z \ (m(x)\longrightarrow s(x,y) \land f(y,z)).$$
What I’m having trouble with is translating to English after negating: 
 I’m having trouble with is translating to English after negating:
∃x ∀y ∃z (m(x)^(¬s(x,y)∨¬f(y,z)))
Is it right if I say some students in this class are not friends with all students in some other faculty
 A: We want to say something about all students in this class:
So, $\forall x (M(x) \to ...$
That is, for all students $x$, if $x$ is a student in this class, then ...
... then there exist(s) some student(s) $y$, such that for all engineering faculty $z,\;F(y, z)$ and $x$ is friends with $y$.
$$\forall x\Big(M(x) \to \exists y \forall z\big(S(x, y) \land F(y, z)\big)\Big)\tag{translation}$$
Your domains are fine.  We use the predicate $M(x)$ to denote the student $x$ is in “this class”, and the predicate $F(y, z)$ to denote $y$ is from the engineering faculty $z$. Note that in the translation, the outermost quantifier quantifies $x$.  
Negating the above, we have
$$\lnot \forall x\Big(M(x) \to \exists y \forall z\big(S(x, y)\land F(y, z)\big)\Big)\tag{$\lnot$ translation}$$
$$\equiv \exists x \lnot\Big(\lnot M(x) \lor \exists y \forall z\big(S(x, y)\land F(y, z)\big)\Big)\tag{$p\to q \equiv \lnot p \lor q$}$$
$$\exists x\Big(M(x) \land \lnot\big(\exists y \forall z( S(x, y)\land F(y, z)\big)\Big)\tag{DeMorgan's}$$
$$\equiv \exists x\Big( M(x) \land \forall y \exists z\big(\lnot S(x,y) \lor \lnot F(y, z)\big)\Big)\tag{DeMorgan's}$$
$$\equiv \exists x\Big(M(x) \land \forall y \big( S(x, y) \to \exists z(\lnot F(y, z))\big)\Big)\tag{$\lnot p \lor \lnot q \equiv p \to \lnot q$}$$
This basically states: There are some students $x$ in this class and all of their friends are not from one or more of the engineering faculty.
