A Parity Pairing Function

For a given positive integer $n$ ($n \geqslant 2$), let $M = \{k \in \mathbb{Z} \mid -n \leqslant k \leqslant n\}$. Nonempty sets $A, B$ satisfy $A \cup B = M$ and $A \cap B = \varnothing$. Suppose that for every $x \in A$ there is a unique $y \in B$ such that $x + y$ is even; write $y = p(x)$ and call $p \colon A \to B$ a $P$-function from $A$ to $B$.

1. For $n = 3$ and $A = \{-3, -1, 1, 3\}$, write down $B$, and decide whether a $P$-function from $A$ to $B$ exists; explain. 2. If $A$ contains at least one odd number and $p \colon A \to B$ is a $P$-function, prove that there exists $x \in A$ with $p(x) = -x$. 3. If $p \colon A \to B$ is a $P$-function such that $p(x) \geqslant x$ for all $x \in A$, find all possible sets $A$.