$8.57 f$ is a one-to-one mapping on $R \rightarrow R$, the function $y=$ $f(x)$ is strictly increasing, the solution set of the equation $x=f(x)$ is $P$, the solution set of the equation $x$ $=f(f(x))$ is $Q$, then
(A) $P \subset Q$.
(B) $P=Q$.
(C) $Q \subset P$.
(D) $P \nsubseteq Q$, and $Q \nsubseteq P$.
(1st "Hope Cup" Mathematics Competition, 1990)
[Solution] If $x \in P$, that is, $x$ satisfies $x=f(x)$,
considering $f(f(x))=f(x)=x$, we know $x \in Q$, thus $P \subset Q$;
Assume $Q \not \subset P$, that is, there exists $x_{0} \in Q$, and $x_{0} \notin P$, such that
$$
f(f(x_{0}))=x_{0}, f(x_{0}) \neq x_{0}.
$$

If $f(x_{0}) < x_{0}$, since $f(x)$ is an increasing function, then $f(f(x_{0})) > f(x_{0})$, which means $x_{0} > f(x_{0})$, a contradiction!
Thus $Q \subset P$. In summary, $P=Q$.
Therefore, the answer is $(B)$.