generated from smyalygames/quartz
44 lines
2.4 KiB
Markdown
44 lines
2.4 KiB
Markdown
---
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lecture: 3
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date: 2025-01-13
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---
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# Proposition 1.1.4
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Each real number is a limit of a sequence of rational numbers.
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$$\mathbb{Q} \subset \mathbb{R}$$
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As an ordered [[Number Field]] $\mathbb{R}$ is **complete**, meaning that every [[Cauchy Sequence]] in $\mathbb{R}$ converges to a real number.
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Equivalently, the real numbers have the **[[Least Upper Bound Property]]**; $\forall X \subset \mathbb{R}$ bounded above has a least upper bound denoted by $sup(X) \in \mathbb{R}$. Eq. an $inf(Y) \in \mathbb{R}$ if $Y$ bounded below.
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# Functions and Cardinality
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A function $f : X \to Y$ is **[[Injective]]** if $f(x) = f(y) \implies x = y$.
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[[Surjective]] if $f(x) = y$.
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[[Bijective]] if it is both injective and surjective.
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Then we write $X \simeq Y$.
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$$|X| = |Y| \; \text{(cardinality)}$$
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Say $X$ is [[Countable]] if $|X| = |\mathbb{N}|$; this means that the members of $X$ can be listed as a sequence with $x_{n} = f(n)$, where $f: \text{IN} \to X$ is some bijection.
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## Cantor's Diagonal Argument
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The real numbers cannot be listed, or they are **uncountable**.
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Indeed, present a list of the real numbers in $\langle 0, 1 \rangle$ written as binary expansions. Then the number that has as its $n$-th digit, the opposite value to the $n$-th digit of the $n$-th number of the list, will never be in the list.
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| **\\** | | | | | | | | | | |
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| ------ | ----- | ----- | ----- | ----- | --- | --- | --- | --- | --- | --- |
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| 0 | **1** | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | ... |
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| 0 | 1 | **1** | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ... |
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| 0 | 0 | 0 | **0** | 0 | 1 | 0 | 0 | 0 | 0 | ... |
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| 0 | 1 | 0 | 1 | **0** | 1 | 0 | 0 | 0 | 0 | ... |
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| ... | | | | | | | | | | |
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So here the bold numbers going diagonally from the **\\** shows that they cannot be countable as they are not the same number.
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Cantor: 0.0011...
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# Axiom of Choice
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Any **[[Direct Product]]**
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$$\Pi_{i \in I} \, X_{i} \equiv \{ x : I \to \cup_{i \in I} X_{i} | x_{i} \equiv x(i) \in X_{i} \}$$
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is non-empty when all $x \neq \emptyset$.
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Any $x \in \Pi_{i \in I} \, X_{i}$ is called a choice function.
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The [[Power Set]] $\wp(X)$ of $X$ consists of all the subsets of $X$.
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$\exists$ bijection $\wp(X) \to \Pi_{x} \{ 0, \, 1 \} = \{ f : X \to \{ 0, \, 1 \} \}$ that sends $Y \subset X$ to its [[Characteristic Function]]. |