3. The set $P$ consisting of certain integers has the following properties: (1) $P$ contains positive and negative numbers; (2) $P$ contains odd and even numbers; (3) $-1 \notin P$; (4) if $x, y \in P$, then $x+y \in P$. Determine the relationship between the real numbers 0 and 2 and the set $P$.
3. Solution From (4) if $x, y \in P$, then $x+y \in P$ we know, if $x \in P$, then $k x \in P(k \in \mathbf{N})$
(1) From (1) we can assume $x, y \in P$, and $x>0, y2 n-1$, then the negative odd number $-2 q m+(2 n-1) \in P$. This leads to a contradiction.