Given a set $A$, denote by $\operatorname{card}(A)$ its cardinal and by $P(A)$ its power set, or set of subsets of $A$.
Suppose we have two sets $A$ and $B$ such that $\operatorname{card}(P(A))\le \operatorname{card}(P(B))$. Can we deduce that $\operatorname{card}(A)\le \operatorname{card}(B)$?
In other more elementary terms, suppose we have an injection $P(A)\hookrightarrow P(B)$. Can we deduce that there is an injection $A\hookrightarrow B$?
Of course I know the answer is yes if some of the sets is finite. But I don't know if it is true in the most easy case that $A$ is denumerable, and $B$ any set.
I know that if the answer is affirmative, then $\operatorname{card}(P(A))= \operatorname{card}(P(B))$ would imply $\operatorname{card}(A)= \operatorname{card}(B)$, a result I don't know if it is true in general.
