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Homework Statement
If A~B, then P(A)~P(B), which means the same as,
If |A|=|B|, then |P(A)|=|P(B)|
Homework Equations
The Attempt at a Solution
This problem is difficult for me because I am trying to learn the methods of proof while at the same time taking Intro Analysis. Anyway, here are my thoughts...
Assume f: A->B is a bijection. From Cantor's Thm. we know that |A|<|P(A)| and |B|<|P(B)|.
I also know that since |A|=|B|, |B|<|P(A)| and |A|<|P(B)|.
After this I feel like saying |P(A)|=|P(B)|, but I feel kind of guilty with that.
Any proof, insight, hint, or a one-liner would be appreciated. Thank you