I'm reviewing for finals and have a sample problem that I think I understand, but would like someone to bless my understanding or smack me and tell me why I'm wrong.
I'm presented with a problem $\Pi$ of unknown complexity class. If I can transform $\Pi$ to some problem $X$, where $X \in {\sf P}$, what does that tell me about $\Pi$?
I think allows me to conclude that $\Pi \in {\sf P}$, right? If I can reduce $\Pi$ to another problem that's deterministically solvable in polynomial time, and the transformation itself can be done "easily" in polynomial time, then I can conclude that $\Pi$ is deterministically solvable in polynomial time, and therefore that $\Pi \in {\sf P}$ correct?
Conversely, given the same input, transforming $X$ to $\Pi$ in polynomial time allows me to conclude nothing meaningful, since nothing is known about $\Pi$ right?