set
A set is a collection of distinct objects which we call elements of the set. A set can have a finite number of elements or infinitely many elements.
declaration of a finite set
We may declare a finite set by listing it's elements between curly braces like so $$\left \lbrace a , b , c , d \right \rbrace$$ or $$\left \lbrace 1 , 2 , 3 , 41 \right \rbrace$$

Suppose we wrote out the set $$\left \lbrace a , b , c , d \right \rbrace$$ and then later on realized that $$a = b = c = d$$, then this set is simply $$\left \lbrace a , a , a , a \right \rbrace$$, which we define to just be $$\left \lbrace a \right \rbrace$$. Therefore we allow duplicates when declaring a set, but realize the set it generates has no such duplicate, so $$\left \lbrace 1 , 1 , 1 \right \rbrace = \left \lbrace 1 , 1 \right \rbrace = \left \lbrace 1 \right \rbrace$$
set builder notation
Suppose that $$P \left ( x \right )$$ represents a statement which depends on the variable $$x$$, then we define $$\left \lbrace x \in S : P \left ( x \right ) \right \rbrace$$ to be the set of elements of $$S$$ such that the statement $$P$$ holds true.
Empty Set
The set with no elements is called the empty set and is denoted by $$\emptyset$$
element of
Suppose that $$a$$ is an element of a set $$S$$, then we write $$a \in S$$
set equality
Given two sets $$A , B$$, we say that $$A$$ and $$B$$ are equal and write $$A = B$$ when $$x \in A$$ if and only if $$x \in B$$
Family of Sets
We denote a set whose elements are also sets by a family of sets to avoid the unwieldy statement: "set of sets"
subset
Given the sets $$A , B$$, we say that $$A$$ is a subset of $$B$$ when for every $$a \in A$$, $$a$$ is also in $$B$$. When this is the case we write $$A \subseteq B$$
superset
Given the sets $$A , B$$, we say that $$A$$ is a superset of $$B$$ when for every $$b \in B$$, $$b$$ is also in $$A$$. When this is the case we write $$A \supseteq B$$
set equality by subsets
Suppose that $$A , B$$ are sets such that $$A \subseteq B$$ and $$B \subseteq A$$, then $$A = B$$

We will show the sets are equal using the definition of set equality.

To show that a bi-implication is true, we have to prove both directions. So first suppose that $$x \in A$$, then since $$A \subseteq B$$, $$x \in B$$. For the other direction suppose that $$x \in B$$, then since $$B \subseteq A$$ then $$x \in A$$. Therefore we know that $$x \in A$$ if and only if $$x \in B$$ as needed.

rationals
The integers are the set $$\mathbb{Q} := \left \lbrace \frac{p}{q} : p , q \in \mathbb{Z} , q \ne 0 \right \rbrace$$

note: The symbol for rationals can be remebered as they are $$\mathbb{Q}$$uotients

integers
The integers are the set $$\mathbb{Z} := \left \lbrace \ldots , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , \ldots \right \rbrace$$

note: The symbol for integers can be remembered as the word for number in German is $$\mathbb{Z}$$ahlen

Natural Numbers 0
We define $$\mathbb{N}_{0} := \left \lbrace 0 , 1 , 2 , 3 , 4 , \ldots \right \rbrace$$
Finite Integer Set
Suppose that $$a, b \in \mathbb{ Z }$$, then the notation $$\left\{ a, \ldots b \right\}$$ is a set $$X$$ such that for any $$x \in \mathbb{ Z }$$ if $$a \le x \le b$$ then $$x \in X$$.
Empty Integer Set
Suppose that $$a, b \in \mathbb{ Z }$$ such that $$a > b$$ then $$\left\{ a, \ldots b \right\} = \emptyset$$
$$\left\{ a, \ldots b \right\}$$ is defined as the set $$X$$ such that $$x \in \mathbb{ X }$$ iff $$x \in \mathbb{ Z }$$ and $$a \le x \le b$$ but this is impossible for if there were any such $$x$$ then by transitivity we have that $$a \le b$$ which is impossible.
Natural Numbers 1
We define $$\mathbb{N}_{1} := \left \lbrace 1 , 2 , 3 , 4 , \ldots \right \rbrace$$
Natural Numbers up to n
Let $$n \in \mathbb{N}_1$$, then we define $$[n] := \{1, 2, \ldots, n\}$$
power set
Given a set $$X$$ we define the power set of subsets $$X$$ as $$P \left ( X \right ) := \left \lbrace S : S \subseteq X \right \rbrace$$