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Given two sets $A,\; B$ and that $|A| = |B|$, show that $|2^A| = |2^B|$.

Intuitively, I think this is true, but I am having trouble showing this formally.

I know that there exists a bijection $f: A \to B$ and that we should try to define a map $g: 2^A \to 2^B$ so that we can show $g$ is bijective. However, I am stuck on defining $g$.

nomly
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  • Hint: an element of $2^A$ is a subset. Apply $g$ to each element. – Kevin Carlson Sep 29 '14 at 17:19
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    I think what @KevinCarlson means is to apply $f$ to each element of the subset of $A$; that is $g(U) = {f(u): u \in U}$ where $U \in 2^A$. – Tom Sep 29 '14 at 17:32
  • Hint: a function is injective if and only if the induced forward-image function is injective. Similarly, a function is surjective if and only if the induced forward-image function is surjective. –  Sep 29 '14 at 17:44
  • @Tom yes, thanks for catching that. – Kevin Carlson Sep 29 '14 at 21:13

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