Negation of a Basic Logic Statement

Question

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:

1. Is my negation correct?

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

Answer 1

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".

Answer 2

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.