generated from smyalygames/quartz
81 lines
3.3 KiB
Markdown
81 lines
3.3 KiB
Markdown
---
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lecture: 1
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date: 2025-01-06
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---
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# Prime Numbers
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$\mathbb{N} = \{1,2,3,\dots\}$ (natural numbers)
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**Prime Number** - $p \in \mathbb{N} \setminus \{1\}$ only divisible by $1$ and $p$.
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They are building blocks for multiplication; for instance
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$90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2$
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$$
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p \times q = p' \times q'
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$$
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$\implies$ (need proof [[#Theorem 1.1.1]])
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$$
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p = p' \; \cap \; q=q'
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$$
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$$
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p = q' \; \cap \; q = p'
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$$
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## Theorem 1.1.1
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Any natural number other than one is a product of unique primes
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### Proof Existence
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Divide as long as possible
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Uniqueness (Gauss): Need Euclid's lemma, saying that
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If $n|ab$ with $gcd(a,b) = 1$, then $n|a$ or $n|b$.
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This lemma follows from the axiom:
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Each non-empty subset of $\mathbb{N}$ has a least element/number.
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[[QED]]
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## COR 1.1.2
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There are infinitely many primes.
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### Proof
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Say we had finitely many primes $p_{1}, \ldots, p_{n}$. Applying [[#Theorem 1.1.1]] to $p_{1} \times p_{2} \times \ldots \times p_{n} + 1$ gives the **absurdity** that $1$ can be divided by some prime number
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This is impossible as for example $p_{1} \times p_{2} \times \ldots \times p_{n} + 1$ and $p_{1} \times p_{7} \times p_{n}$, you can divide both sides by something like $p_{1}$, however on the LHS with $+ 1$ would result in $+ 1 \frac{1}{p_{1}}$
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QED
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## Statements
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These are mostly similar to logic in computer science with [[And]], [[Or]], [[Not]], and [[Implies]].
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# Sets
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A **set** $X$ is characterised by its **elements** or **members** $x \in X$.
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They can be listed like $\{1,5,4\}$, or described by some property, like the set of all primes, or like $X = \{x | P(x)\}$; here $x$ is from the outset supposed to belong to some (universal) set. Otherwise $X = \{ x | \notin X \}$ (Russel's paradox) - which is not allowed. $X = \{ x \in | x > 7\}$ - which is OK!
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$Y \subset$ means $x \in Y \implies x \in X$
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Get $\emptyset \subset X$
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## Union
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The **union** $\cup_{i \in I} X_{i}$ consists of $x \in X_{i}$ for at least one $i \in I$
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**Disjoint** union when $X_{i} \cap X_{j} = \emptyset$ for all possible $i$ and $j$.
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## Intersection
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The **intersection** $\cap_{i \in I} X_{i}$ consists of $x \in X_{i}, \; \forall i \in I$
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## Complement
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The **complement** $X \setminus Y$ of $Y$ in $X$ consists of $x \in X \cap x \notin Y$
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Write $Y^\complement$ (a complement of $Y$) when $X$ is understood.
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## Product
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The **product** $X \times Y$ consists of the **ordered pairs**
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$$(x, y) \neq (y, x) \equiv \{ \{ y \}, \{ y, x \} \}$$
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($x \in X$ and $y \in Y$)
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$$
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(x,y) = (x', y') \iff x = x' \cap y = y'
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$$
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A more compact way of writing this: $X \times Y = \{ (x, y) | x \in X \cap y \in Y \}$
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## Relation
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A **relation** on a set $X$ is $R \subset X \times X$ with $xRy \equiv ((x,y) \in R)$. ($R$ here meaning is related to)
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### Example
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$R= \{ (x, x) | x \in X \} \subset X \times X$, $xRy \iff (x, y) \in R \implies x = y$.
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These two elements $x$ and $y$ can only relate if they are the same.
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### Equivalence Relation
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An **equivalence relation** $0 \sim X$ is a relation on $\sim$ on $X$ such that:
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1. $x \sim x$
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2. $x \sim y \implies y \sim x$
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3. $(x \sim y) \cap (y \sim z) \implies x \sim z$
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For all $x,y,z \in X$.
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It **partitions** $X$ into a disjoint union $\frac{X}{\sim}$ of **equivalence classes** $[x] \equiv \{ y \in X | y \sim x \}$, with $x$ called a **representative** of $[x]$ (equivalence class). |