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 { a , b , c , d } or { 1 , 2 , 3 , 41 }

Suppose we wrote out the set { a , b , c , d } and then later on realized that a = b = c = d , then this set is simply { a , a , a , a } , which we define to just be { a } . Therefore we allow duplicates when declaring a set, but realize the set it generates has no such duplicate, so { 1 , 1 , 1 } = { 1 , 1 } = { 1 }
set builder notation
Suppose that P ( x ) represents a statement which depends on the variable x , then we define { x ∈ S : P ( x ) } 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 ∅
element of
Suppose that a is an element of a set S , then we write a ∈ S
set equality
Given two sets A , B , we say that A and B are equal and write A = B when x ∈ A if and only if x ∈ 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 ∈ A , a is also in B . When this is the case we write A ⊆ B
superset
Given the sets A , B , we say that A is a superset of B when for every b ∈ B , b is also in A . When this is the case we write A ⊇ B
set equality by subsets
Suppose that A , B are sets such that A ⊆ B and B ⊆ 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 ∈ A , then since A ⊆ B , x ∈ B . For the other direction suppose that x ∈ B , then since B ⊆ A then x ∈ A . Therefore we know that x ∈ A if and only if x ∈ B as needed.

rationals
The integers are the set Q := { p q : p , q ∈ Z , q ≠ 0 }

note: The symbol for rationals can be remebered as they are Q uotients

integers
The integers are the set Z := { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … }

note: The symbol for integers can be remembered as the word for number in German is Z ahlen

Natural Numbers 0
We define N 0 := { 0 , 1 , 2 , 3 , 4 , … }
Finite Integer Set
Suppose that a , b ∈ Z , then the notation { a , … b } is a set X such that for any x ∈ Z if a ≤ x ≤ b then x ∈ X .
Empty Integer Set
Suppose that a , b ∈ Z such that a > b then { a , … b } = ∅
{ a , … b } is defined as the set X such that x ∈ X iff x ∈ Z and a ≤ x ≤ b but this is impossible for if there were any such x then by transitivity we have that a ≤ b which is impossible.
Natural Numbers 1
We define N 1 := { 1 , 2 , 3 , 4 , … }
Natural Numbers up to n
Let n ∈ N 1 , then we define [ n ] := { 1 , 2 , … , n }
power set
Given a set X we define the power set of subsets X as P ( X ) := { S : S ⊆ X }