Negation of a statement without using (symbol ¬) 
Write the negation of the following statement (without using the symbol $¬$ ):
$\mathrm P ~=~ (∃x ∈ \Bbb R)\Big(\big((∃y ∈ \Bbb R)(x = (1 − y)^2)\big) ∧ \big((∃z ∈ \Bbb R)(x = −z^2)\big)\Big)$
Which statement is true, $\rm P$ or $\rm ¬P$ ?

P.S.... i'm kinda confused as to not being able to use the $¬$ symbol for negating the statement.   So, how would I be able to do it?

What you are supposed to do, is the following: Consider $\neg P$ and then iteratively transform this into an equivalent sentence $Q$ in which the symbols $\neg$ doesn't appear. I will show you how to begin:
(I'm assuming that your $R$ is supposed to be $\mathbb R$ - the set of all reals.)
$$
\begin{align}
&\neg \left( \exists x \in \mathbb R \exists y \in \mathbb R \colon x = (1-y)^2 \wedge \exists z \in \mathbb R \colon x = -z^2 \right) \\
\Leftrightarrow &\neg \left( \exists x \in \mathbb R \exists y \in \mathbb R \colon x = (1-y)^2 \right) \vee \neg \left( \exists z \in \mathbb R \colon x = -z^2 \right) \\
\Leftrightarrow &\forall x \in \mathbb R \neg \left( \exists y \in \mathbb R \colon x = (1-y)^2 \right) \vee \forall z \in \mathbb R \colon x \neq -z^2 \\
&\ldots
\end{align}
$$
