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I am not familiar with all rule but I read about infinite set and Cantor's diagonal. I want to prove there is one to one correspondence between power set of natural number and real number. I search a lot but I don't find anything. Is it possible to give me a reference or prove that for me here.
I'm sorry for bad English.
Thanks.

Amin
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1 Answers1

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There is a bijection between $P(\mathbb{N})$ And $(0, 1)$. Consider the numbers represented in binary after the point sign. Then if your sub set of naturals contains $n$ you set the $n+1$ didgit to 1 and to 0 otherwise. This gives a unique encoding of each subset in the reals and vise versa.

For example $\{1\}$ is mapped to $0.01$, $\{1,2\}$ is mapped to $0.011$ and $\{3\}$ gives $0.0001$.

Q the Platypus
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  • can you explain more with example? for example I relate {1} to ... or relate {1, 2} to ... or relate {3} to .... can you explain for me for this? – Amin Mar 10 '18 at 16:53