Cantor's Theorem: for any set S, |\mathcal{P}(S)| > |S|
for finite set, |\mathcal{P}(S)|=2^n
for infinite set, we have a chain: |\mathbb{N}|<|\mathcal{P}(\mathbb{N})|<|\mathcal{P}(\mathcal{P}(\mathbb{N}))|<...
Proof: we construct a set that has all the input such that we can't find input itself in the P(input)
TODO: I don't understandConcrete Proof
Theorem: \{x \in \mathbb{R} | 0 < x < 1\} is uncountable.
Proof: there exists no N->(0,1) bijection
first construct a generic f(x) representation of the mapping
then make a number in (0, 1) that has not been mapped to by any other x
f is not subjective
done
Corollary: |(0, 1)| = |\mathbb{R}|
Irrational Numbers (\mathbb{I}) is uncountable.
Uncountable sets: (use without proof)
\mathcal{P}(\mathbb{N})
\mathbb{R}
(0, 1)
\{0, 1\}^{\mathbb{R}} (infinite zeros and ones)
Countable sets:
\mathbb{Z}
\mathbb{N}
\mathbb{Q}
Think \mathcal{P}(\mathbb{N}) as binary representation. and you can use \mathbb{R} with 0.0101010010. to prove.
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