How can I prove that if $f: A \rightarrow B$ is bijective then $g: P(A) \rightarrow P(B)$ is also bijective. $P(A)$ and $P(B)$ are power sets.
For injectivity, is it a sufficient proof to say that: Suppose $X_1,X_2\in P(B)$ and $f(X_1)=f(X_2)$. Since $f$ is injective since it is bijective we can see that $X_1=X_2$?
Don't know how to start surjectivity proof.