---
lecture: 1
date: 2025-01-06
---
# Prime Numbers
$\mathbb{N} = \{1,2,3,\dots\}$ (natural numbers)

**Prime Number** - $p \in \mathbb{N} \setminus \{1\}$ only divisible by $1$ and $p$.

They are building blocks for multiplication; for instance
$90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = 2 \times 3^2 \times 5 = 5 \times 3^2 \times 2$

$$
p \times q = p' \times q'
$$
$\implies$ (need proof [[#Theorem 1.1.1]])
$$
p = p' \; \cap \; q=q'
$$
$$
p = q' \; \cap \; q = p'
$$
## Theorem 1.1.1
Any natural number other than one is a product of unique primes
### Proof Existence
Divide as long as possible
Uniqueness (Gauss): Need Euclid's lemma, saying that
If $n|ab$ with $gcd(a,b) = 1$, then $n|a$ or $n|b$.

This lemma follows from the axiom:
Each non-empty subset of $\mathbb{N}$ has a least element/number.

[[QED]]
## COR 1.1.2
There are infinitely many primes.
### Proof
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

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 $+ 1 \frac{1}{p_{1}}$

QED
## Statements
These are mostly similar to logic in computer science with [[And]], [[Or]], [[Not]], and [[Implies]].
# Sets
A **set** $X$ is characterised by its **elements** or **members** $x \in X$.

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!

$Y \subset$ means $x \in Y \implies x \in X$

Get $\emptyset \subset X$
## Union
The **union** $\cup_{i \in I} X_{i}$ consists of $x \in X_{i}$ for at least one $i \in I$

**Disjoint** union when $X_{i} \cap X_{j} = \emptyset$ for all possible $i$ and $j$.
## Intersection
The **intersection** $\cap_{i \in I} X_{i}$ consists of $x \in X_{i}, \; \forall i \in I$
## Complement
The **complement** $X \setminus Y$ of $Y$ in $X$ consists of $x \in X \cap x \notin Y$
Write $Y^\complement$ (a complement of $Y$) when $X$ is understood.
## Product
The **product** $X \times Y$ consists of the **ordered pairs**
$$(x, y) \neq (y, x) \equiv \{ \{ y \}, \{ y, x \} \}$$
($x \in X$ and $y \in Y$)
$$
(x,y) = (x', y') \iff x = x' \cap y = y'
$$
A more compact way of writing this: $X \times Y = \{ (x, y) | x \in X \cap y \in Y \}$
## Relation
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)
### Example
$R= \{ (x, x) | x \in X \} \subset X \times X$, $xRy \iff (x, y) \in R \implies x = y$.
These two elements $x$ and $y$ can only relate if they are the same.
### Equivalence Relation
An **equivalence relation** $0 \sim X$ is a relation on $\sim$ on $X$ such that:
1. $x \sim x$
2. $x \sim y \implies y \sim x$
3. $(x \sim y) \cap (y \sim z) \implies x \sim z$
For all $x,y,z \in X$.

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).