Just started studying set theory. It's seems to me intuitivly correct that if $A<_c B \implies P(A)<_c P(B)$ where $_c$ is the cardinality of a set and $P(\cdot)$ is the powerset. Am I right? I got confused of which injection to take from $P(A)\to P(B)$. Need some help.
Thank you.
No, this is not true.
It is consistent that $2^{\aleph_0}=2^{\aleph_1}=\aleph_2$. In that case you don't have the sharp inequality on the right hand side. What is true is that you always have: $$A\leq_c B\implies \mathcal P(A)\leq_c\mathcal P(B).$$