how to show that sets $\{0,1\}^{\mathbb{N}}$ and $\mathbb{R}^{\mathbb{N}}$ have an equal cardinality?
I tried to use Cantor-Bernstein theorem but it seems to be hard.
Sets of equal cardinality
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elementary-set-theory
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0Hint: Do it in two steps: first show that $\{0,1\}^{\mathbb{N}}$ and $\mathbb{R}$ have the same cardinality, then that $\mathbb{R}$ and $\mathbb{R}^{\mathbb{N}}$ also share the same cardinality. Conclude by transitivity. – 2017-01-07
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0Hi, i'm trying to show that $\mathbb{R}$ have the same cardinality as $\mathbb{R}^{\mathbb{N}}$ – 2017-01-07
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0@JohnCyna Note that $\{0,1\}^{\Bbb N}$ and $\{0,1\}^{\Bbb N\times \Bbb N}$ have same cardinality because $\Bbb N$ and $\Bbb N\times \Bbb R$ have – 2017-01-07
1 Answers
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You may replace $\Bbb R$ with $[0,1)$. The elements of $[0,1)$ are represented as infinite binary sequences. You can get from sequences of sequences of binary digits to sequences of binary digits by a zig-zag-enumaration as in the diagonal argument.