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.

Oh I meant 1/(2x)! Sorry about that, still getting used to this formatting. — MarsIsPrettyClose, Sep 20 '14 at 19:41

score 0 · Answer 1 · answered Sep 20 '14 at 19:35

0

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

answered Sep 20 '14 at 19:35

Aaron Maroja

17,571

Is the last part not negated? Why is that? – MarsIsPrettyClose Sep 20 '14 at 19:37
I was looking to your negation and got it wrong, just edited now – Aaron Maroja Sep 20 '14 at 19:39
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Oh that makes more sense, thank you so much! That clarified everything, I've been stuck on 10+ questions because I didn't know you can negate x^2 in N in such way. Thanks again. :) – MarsIsPrettyClose Sep 20 '14 at 19:40
Just remember you're not negating x$^2 \in N$ and rather the sentence, that is, $\forall p \in N, P \wedge \sim Q$, which is $\exists p \in N; \sim P \vee Q$. – Aaron Maroja Sep 20 '14 at 19:44
I'm glad I could help – Aaron Maroja Sep 20 '14 at 19:45

score 0 · Accepted Answer · answered Sep 20 '14 at 19:35

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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.

answered Sep 20 '14 at 19:35

karakusc

1,512

Than you so much, I did not know negate x^2 in N that way!! Very helpful. :) – MarsIsPrettyClose Sep 20 '14 at 19:39

score 0 · Answer 3 · answered Sep 20 '14 at 19:42

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.

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

Negation of this statement

3 Answers3