1. ## Negation Question

In my discrete math class I was given the following problem

Let A and B be sets of real numbers. Write the negation of
each of the following statements:
(a) For every a ∈ A, it is true that a^2 ∈ B.
(b) For at least one a∈ A,it is true that a^2 ∈ B.

Would the negation for each question be correct if I said:

For every a that is not an element of A it is not true that a^2 is not an element of B

and

For at least one a that is not an element of A it is not that that a^2 is not an element of B

ANy help provided would be appreciated. Also does anyone know any good books for discrete math to use as a reference? The class I am in does not have a required textbook we just work off your instructors notes.

Thanks
and

2. This is the negation of (a): $\left( {\exists x \in A} \right)\left[ {x^2 \notin B} \right]$

3. Consider a statement $\mathcal{Q}_1x_1\in A_1\dots\mathcal{Q}_nx_n\in A_n\,P(x_1,\dots,x_n)$ where each $\mathcal{Q}_i$ is either $\forall$ or $\exists$. Then its negation is $\bar{\mathcal{Q}}_1x_1\in A_1\dots\bar{\mathcal{Q}}_nx_n\in A_n\,\neg P(x_1,\dots,x_n)$, where $\bar{\exists}$ is $\forall$ and $\bar{\forall}$ is $\exists$.