70 lines
1.8 KiB
Markdown
70 lines
1.8 KiB
Markdown
# Math4121 Lecture 17
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## Continue on Last lecture
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### Countability
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#### Theorem: $\mathbb{R}$ is uncountable
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We denote the cardinality of $\mathbb{N}$ be $\aleph_0$
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We denote the cardinality of $\mathbb{R}$ be $\mathfrak{c}$
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> Continuum Hypothesis:
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>
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> If there a cardinality between $\aleph_0$ and $\mathfrak{c}$
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### Power set
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#### Definition: Power set
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Given a set $S$, the power set of $S$, denoted $\mathscr{P}(S)$ or $2^S$, is the collection of all subsets of $S$.
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#### Theorem 3.10 (Cantor's Theorem)
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Cardinality of $2^S$ is not equal to the cardinality of $S$.
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<details>
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<summary>Proof of Cantor's Theorem</summary>
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Assume they have the same cardinality, then $\exists \psi: S \to 2^X$ which is one-to-one and onto. (this function returns a subset of $S$)
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$$
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T=\{a\in S:a\notin \psi (a)\}\subseteq S
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$$
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Thus, $\exists b\in S$ such that $\psi(b)=T$.
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If $b\in T$, then by definition of $T$, $b \notin \psi(b)$, but $\psi(b) = T$, which is a contradiction. So $b\notin T$.
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If $b \notin T$, then $b \in \psi(b)$, which is also a contradiction since $b\in T$. Therefore, $2^S$ cannot have the same cardinality as $S$.
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</details>
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### Back to Hankel's Conjecture
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$$
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T=\bigcup_{n=1}^\infty \left(a_n-\frac{\epsilon}{2^{n+1}},a_n+\frac{\epsilon}{2^{n+1}}\right)
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$$
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is small
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What is the structure of $S=[0,1]\setminus T$? (or Sparse)
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- Cardinality (countable)
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- Topologically (not dense)
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- Measure, for now meaning small or zero outer content.
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## Chapter 4: Nowhere Dense Sets and the Problem with the Fundamental Theorem of Calculus
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### Nowhere Dense Sets
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#### Definition: Nowhere Dense Set
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A set $S$ is **nowhere dense** if there are no open intervals in which $S$ is dense.
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#### Corollary: A set is nowhere dense if and only if $S$ contains no open intervals
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$S'$ contains no open intervals
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