Abstract Algebra

https://math.berkeley.edu/~apaulin/AbstractAlgebra.pdf
These are my notes from this university pdf notes

Defining Unity

The concept of Unity. The number $1$ , or $I$ or $e$ ... All represents the same thing.
Let $B$ be the set of numbers. And the operator be defined as $*$ . If they satisfy 4 properties, they form a Group.

(B, *) \to Group

The unity $e \in B$ must satisfy $e * a = a, \forall a \in B$ .

Set Theory basics

Notation

Let $S$ and $T$ be two sets

$S = {Notation for elements in S | Properties which specifies being in S}$
The vertical bar should be read as "such that"

Complementary Set

If $S \in T$ , then $T ∖ S := {x \in T | x \notin S}$ is called the compliment of $S$ in $T$ .

Cartesian Product

$S \times T = {(a, b) | a \in S, b \in T}$ is the cartesian product of $S$ and $T$ .

Notation

Notation	Meaning
$a \in G$	$a$ in $G$
$\forall a$	for all values of $a$
$\exists a$	there exists $a$
$\exists! a$	there exists unique $a$
$\sum_{i = a}^{b} i$	summation of i from a to b
$⋃_{i = a}^{b} S_{i}$	the union of all the sets $S_{a}$ to $S_{b}$
$lim_{x \to a} f (x)$	the value $f (x)$ approaches as $x$ approaches $a$
$sup_{x \in [a, b]} f (x)$	the smallest upper bound of $f (x)$ over $x \in [a, b]$
$max_{x \in [a, b]} f (x)$	the maximum of $f (x)$ over $x \in [a, b]$
$∴$	Therefore
$∵$	Because
$x \mapsto y$	Elements from $x$ maps to $y$
$f : S \to T$	function maps elements from $S$ to $T$
$a ⟹ b$	$a$ implies $b$ (if $a$ is true, $b$ is true)
$a ⟺ b$	$a$ is true if and only if $b$ is true

Functions

Definition

A map (or function) $f$ from $S$ to $T$ is a rule which assigns element of $S$ a unique elements of $T$ .

f : S \to T

x \mapsto f (x)

Eg. $f : Z \times Z \to Z$ can be a map from $(a, b) \mapsto a + b$ .

In the case $f : S \to T$

$S$ is the domain of $f$ and $T$ is it's codomain (not the same as range)
$f$ is the identity map if $f (x) = x, \forall x \in S$ and $S = T$ . We can rewrite it as $f = I d_{S}$
$f$ is Injective if $f (x) = f (y) ⟹ x = y, \forall x, y \in S$ . One to one.
$f$ is Surjective if $y \in T, x \in S, \exists x f (x) = y$ . All $T$ is mapped
$f$ is bijective if it's both Injective and Surjective.

Equivalence Relation

The notation $x \sim y$ represents " $x$ is equivalent to $y$ ". Suppose we define equivalence as numbers which has the same parity (both even or odd). Then, if $S = {1, 2, 3, 4}$ , $2$ is equivalent to $4$ as they are both even. $∴ 2 \sim 4$ ...

Let $U$ be a relation on set $S$ . $U \subseteq S \times S$ where $x \sim y ⟺ (x, y) \in U$ where $x, y \in S$ .

Symmetry Property

x \sim y ⟹ y \sim x

(x, y) \in U ⟹ (y, x) \in U

Reflexive Property

x \sim x, \forall x \in S

Transitive Property

Given $x, y, z \in S$ .

(x, y) \in U and (y, z) \in U ⟹ (x, z) \in U

Equivalence class

The equivalence class of $x$ is the set of all elements in $S$ that are "equivalent" to $x$ .

[x] := {y \in S ∣ y \sim x} \subset S

$[2] = {2, 4}$ and $[1] = {1, 3}$ are examples.

Remarks

Implies that

$x \in [x]$ , hence $[x] \neq \emptyset$
$y \in [x] ⟺ [y] = [x]$ . Hence two equivalent classes are either equal or disjoint.

Partitions

A partition ${X_{i}}$ of set $S$ is defined such that its union is $S$ , $⋃_{i \in I} X_{i} = S$ , none of its elements is empty, $X_{i \in I} \neq \emptyset$ and it is pairwise disjoint, $X_{i} \cap X_{j} = \emptyset \forall i, j \in I i \neq j$ . Notice how our equivalence class are partitions of set $S$ .

The study of $Z$

We can express addition and multiplication as such

+ : Z \times Z \to Z

(a, b) \mapsto a + b

\times : Z \times Z \to Z

(a, b) \mapsto a \times b

Recall Group Theory,

Defining Unity

Set Theory basics

Notation

Complementary Set

Cartesian Product

Notation

Functions

Equivalence Relation

Symmetry Property

Reflexive Property

Transitive Property

Equivalence class

Remarks

Partitions

The study of Z

The study of $Z$