\[
\displaylines{\text{Let }X\text{ be a set, and let }A\subseteq X.\text{ Define a function }f:P\left(x\right)\to P\left(X\right)\text{ by }f\left(B\right)=A\cap B,\text{ for }B\in P\left(X\right).\\ \text{Under what conditions is }f\text{ one-to-one and onto?}}
\]