Negation of a Basic Logic Statement I am trying to negate the following english statement: All manatees are swimming.
m(x): x is a manatee
s(x): x is swimming

$\therefore (\forall x)(m(x) \land s(x))$
\begin{align*}
\neg [(\forall x)(m(x) \land s(x))] &\equiv \\
(\exists x) \neg(m(x) \land s(x)) &\equiv & \textbf{DeMorgan's Law}\\
\boxed{(\exists x) (\neg m(x) \lor \neg s(x))}
\end{align*}
The negation: Some sea-creatures are not manatees or not swimming.
My questions are:


*

*Is my negation correct? 

*Can I infer that the domain is "sea-creatures" or do I need to say "there are things"**?
 A: 
Is my negation correct?

Yes, your negation is correct
However, note that all manatees are swimming  translate to:
$$\forall x(m(x)\to s(x))$$
Negation is $\exists x(m(x)\land \neg s(x))$ translate to some meanatee is not swimming.
$\forall x(m(x)\land s(x))$ means everything (in the domain of $x$) is swimming meanatee.

Can I infer that the domain is "sea-creatures" or do I need to say "there are things"?

Since on the first part it says $\forall x(m(x)\land s(x))$ means all manatees are swimming which is equivalent to $\forall x(m(x)\to s(x))$, that happens only if the domain of $x$ is all meanatees, that we can prove this two statement are the same:
\begin{align}
&\forall x(m(x)\land s(x))\\
\equiv&\forall x(\top\land s(x))\\
\equiv&\forall x(s(x))\\
\end{align}
\begin{align}
&\forall x(m(x)\to s(x))\\
\equiv&\forall x(\top\to s(x))\\
\equiv&\forall x(s(x))\\
\end{align}
Hence $\forall x(m(x)\land s(x))\equiv\forall x(m(x)\to s(x))$ when domain is all meanatees.
A: Your translation of the original statement is false. It should be $\forall x:m(x)\to s(x)$ or $\forall x:\neg m(x)\lor s(x)$. Then its negation is $\exists x:m(x)\land\neg s(x)$, or "there exists a manatee that is not swimming".
