Negating statements I have midterm exam tomorrow, and I am having trouble in negating statements. 
Am I right? I have to negate the following statements:


*

*For every positive integer a, there exists an integer b with $|b| < a$ such that b divides a.

*For all n belongs to Integer, there exists an integer k ≠ 2 such that $k|(n^3-n)$

*For every integer m such that $2|m$ and $4∤m$,there are no integers x and y that
satisfy $x^2 + 3y^2 = m$.


I have the following:


*

*There exist a positive integer a such that for all integer b with $|b| < a$, b divides a.

*There exist an integer n such that for all integer k, except 2, $k∤(n^3-n)$

*There exist integer m with $2|m$ and $4∤m$, such that there are integers x and y that satisfy $x^2 + 3y^2 = m$.

 A: For 1. : 

For every positive integer a, there exists an integer b with $|b|<a$ such that b divides a --- (*)

we have in symbols :

$\forall a (a > 0 \rightarrow \exists b (|b|<a \land b|a))$.

Negating it, we have to "reverse" the quantifiers (as you have done), but we have not to forget the negation! (and use the fact that $p \rightarrow q$ is equivalent to $\lnot p \lor q$ and De Morgan) :

$\exists a (a > 0 \land \forall b \lnot (|b|<a \land b|a))$.

Finally, we can rewrite the last formula as :

$\exists a (a > 0 \land \forall b (|b| < a \rightarrow \lnot b|a))$.

Thus the negation of (*) is :


There exists a positive integer a such that, for every integer b with $|b|<a$, b does not divide a. 



For 2. :

For all n belongs to integer, there exists an integer $k \ne 2$ such that $k|(n^3−n)$ ---(§)

we have :

$\forall n \exists k (k \ne 2 \land k|(n^3−n))$.

Thus the (correct) negation is :

$\exists n \forall k \lnot (k \ne 2 \land k|(n^3−n))$

i.e.

$\exists n \forall k (k \ne 2 \rightarrow  \lnot k|(n^3−n))$.

Thus the negation of (§) is :


There exists an integer n such that, for every integer k different from 2, k does not divide $(n^3−n)$. 


