If cardinality of set $A$ is less or equal to the cardinality of set $B$ then cardinality of the power set of $A$ is less or equal to the cardinality of power set of $B$
This holds for finite sets, i want to know if it does for infinte ones too. I tried to do a proof:
Since $|A|\le |B|$ then there exist a one to one function $f: A \to B$
To show $ |P(A)|\le|P(B)| $ we have to find a one to one function $g: P(A) \to P(B)$
Define $g$ such that it maps $\{a_k\}~\to \{f(a_k)\}$
Likewise $\{a_j,~...~,~a_n\}~\to \{f(a_j),~...~,~f(a_n)\}$.
$g$ is injective because $f$ is. I am not sure if $g$ is well defined for infinte subsets ?