Lecture 021

Uncountability

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: $|\mathbb{N}|<|\mathcal{P}(\mathbb{N})|<|\mathcal{P}(\mathcal{P}(\mathbb{N}))|<...$

Diagonazation

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 understand

Concrete Proof

Theorem: $\{x \in \mathbb{R} | 0 < x < 1\}$ is uncountable.

• Side note: for $x\in \mathbb{R}$, decimal representation is not unique. For example: 0.129999...=0.130000... Therefore we chose to always represent numbers with trailing zeros

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

Extra Memorization

Corollary: $|(0, 1)| = |\mathbb{R}|$

• as well as (a, b), (a, b]...[a, \inf]

Irrational Numbers ($\mathbb{I}$) is uncountable.

• otherwise $\mathbb{R} = \mathbb{I} \cup \mathbb{Q}$ is countable. -> contradiction

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.