P-Completeness and Reducibility I am taking an algorithm analysis class and am stuck on one of my homework problems and would appreciate it if I could receive some guidance.

The problem I'm stuck on is proving that the empty language and $\{0, 1\}^*$ are the only languages in P that are not complete for P with respect to polynomial-time reductions (problem 34.3-6 in CLRS 3rd edition). The first part of the problem seems fairly straightforward enough (proving the empty language criteria). However, I'm not sure where to even begin when I have to prove the criteria for $\{0, 1\}^*$. I'm NOT looking for the answer, however I would appreciate some guidance on how I can begin to think about this problem.
Hint. The definition of reduction from $X$ to&nbsp;$Y$ requires that "yes" instances of&nbsp;$X$ be mapped to "yes" instances of&nbsp;$Y$ and "no" instances to "no" instances.