1. ## Negations Project

Hey all, I need to negate the following statements, can I get feedback on if this is correct?

Every cubic polynomial has a real root.
Negation: There exists a cubic polynomial that has no real root.

G is normal and H is regular.
Negation: G is not normal or H is not regular.

There does not exist 0 such that for all x, x+0=x
Negation: For no 0 does there exist x, where x+0 =/ x

The newspaper article was neither accurate nor entertaining.
Negation: The newspaper article was accurate and entertaining.

If gcd(m,n) is odd, then m or n is odd.
Negation: Gcd(m,n) is odd and m is not odd and n is not odd.

H/N is a normal subgroup of G/N if and only if H is a normal subgroup of G.
Negation: H/N is a normal subgroup of G/N and H is not a normal subgroup of G or H is a normal subgroup of G and H/N is not a normal subgroup of G/N.

For each epsilon > 0 there exists N in the Natural Numbers such that for all n >= N, the abs|An - L| < epsilon

Negation: There exists epsilon > 0 for each N in the Natural Numbers such that there exists n >= N, |An - L| >= epsilon

Any help would be appreciated!

2. There does not exist 0 such that for all x, x+0=x
Negation: For no 0 does there exist x, where x+0 =/ x
The original statement is equivalent to "It is not the case that there exists 0 such that for all x, x+0=x", so, to form negation, you just remove "It is not the case that".

The newspaper article was neither accurate nor entertaining.
Negation: The newspaper article was accurate and entertaining.
Should be "accurate or entertaining".

For each epsilon > 0 there exists N in the Natural Numbers such that for all n >= N, the abs|An - L| < epsilon

Negation: There exists epsilon > 0 for each N in the Natural Numbers such that there exists n >= N, |An - L| >= epsilon
You are making this more complicated than necessary. "There exists epsilon > 0 for each N" means "For each N there exists an epsilon > 0 (which may depend on N)", and it's a wrong way to start the negation. Rather, you keep the same order of variables: epslion, N, n, but change the quantifiers into the dual ones. "There exists an epsilon > 0 such that for each N in the Natural Numbers, there exists an n >= N such that |An - L| >= epsilon".

The rest seems good to me.