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**1. Homework Statement**

Let A and B be infinite sets with the same cardinality. Prove that P(A) and P(B) have the same cardinality. Do this by giving

*explicitly*a bijective function from P(A) to P(B). You must also prove that your function is indeed a bijection.

**2. Homework Equations**

**3. The Attempt at a Solution**

To be honest, I have absolutely no idea on how to even approach this problem. After the 3+ hours of Office hours (bless my T.A's heart, being patient with me and all), all I could come up with is:

We know that g:A -> B is a bijection and we have to show that f:P(A) ->P(B) is a bijection. I also know that I have to somehow find this bijective function for P(A) and P(B), and when I do that, I have show that the function is one-to-one and onto.

So since g:A->B is bijective we know we could have something like

g:

a1 ---> b1

a2 ---> b2 and so on.

My T.A. also wrote down:

{a} ---> {b}

{a1, a2} ---> {b1, b2}

{b2} ---> {b2}

Which he said could possibly help in finding the bijective function (which he said should probably be in set builder form). Any help would be greatly appreciated.