My proofs professor gave us this problem as a challenge, but I'm stuck, which is why I'm here:(adsbygoogle = window.adsbygoogle || []).push({});

Let A,B sets

Define set A to be (0,1)

Define set B to be P(N), where N is the set of natural numbers, and P(N) is its power set.

Question: Construct a bijection between (0,1) and P(N) or a related set.

So I need to show that [tex] \exists f:A \rightarrow B[/tex], a bijection

Now, I know that both (0,1) and P(N) are both uncountably infinite.

The first thought that came to me was to use the Shroeder-Bernstein theorem and prove that set A is 1:1 with set B and vice verse, which would make it onto as well.

But, I guess, what I'm failing to grasp, is how an uncountably infinite set can be 1:1 with another uncountably infinite set, and vice verse, and thus be a bijective mapping?

He did also say that we could try to prove that both sets have the same cardinality, but did not elaborate. He mentioned that this was going to be difficult. I guess I'm just having difficulty starting.

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# Proofs question

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