Negation of this statement

Question

Having some trouble negating this expression

\begin{align} \forall x \in N, x^2 \in N \text { and } (1 / 2x) \notin N \end{align}

"Indicate whether the negated statement is true."

When I read it out loud it makes sense to me.

"For all x in the natural numbers, x^2 is also in the natural numbers (which makes sense) and 1 / 2x is not in the natural numbers (which also makes sense).

My go at negating this

\begin{align} \lnot \forall x \in N, x^2 \in \text { and } (1 / 2x) \in N \end{align}

Is this correct? Seems wrong to me.

The question also asks not to keep the negation symbol at the front, which is confusing me even more.

Any tips would be appreciated, thank you.

Answer 1

When you negate a logical expression, you convert $\forall$'s to $\exists$'s, and ANDs to ORs (and vice versa). So the negation would be

$$ \exists x \in N, x^2 \notin N \text{ or } (1/2x) \in N $$

So assuming N denotes the natural numbers, the negation is not true, since you cannot find any natural number satisfying either of the predicates.

Answer 2

$\exists x \in N$ sucht that $x^2 \notin N$ or $\frac{1}{2x} \in N$.

Answer 3

Indicate whether the negated statement is true.

Assuming that $1/2x$ means $\frac{1}{2}x$, $4 \in \mathbb{N}$ and $\frac{1}{2}(4)=2 \in \mathbb{N},$ in which case the statement is false.

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If the $1/2x$ means $\frac{1}{2x}$, then the statement is true.