Lecture 1 - 1.1 Sets and Numbers

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
$90=9\times 10=3\times 3\times 2\times 5=2\times 3^2\times 5=5\times 3^2\times 2$

$$p\times q=p^{\prime }\times q^{\prime }$$

$⟹$ (need proof Theorem 1.1.1)

$$p=p^{\prime }\cap q=q^{\prime }$$ $$p=q^{\prime }\cap q=p^{\prime }$$

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 $p_1,\dots ,p_n$. Applying Theorem 1.1.1 to $p_1\times p_2\times \dots \times p_n+1$ gives the absurdity that $1$ can be divided by some prime number

This is impossible as for example $p_1\times p_2\times \dots \times p_n+1$ 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 $x\in Y⟹x\in X$

Get $\varnothing \subset X$

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 $X_i\cap X_j=\varnothing$ 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
$(x,y)\ne (y,x)\equiv \{\{y\},\{y,x\}\}$
($x\in X$ and $y\in Y$)

$$(x,y)=(x^{\prime },y^{\prime })⟺x=x^{\prime }\cap y=y^{\prime }$$

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

$R=\{(x,x)|x\in X\}\subset X\times X$, $xRy⟺(x,y)\in R⟹x=y$.
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. $x\sim y⟹y\sim x$
3. $(x\sim y)\cap (y\sim z)⟹x\sim z$
   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).