ΘρϵηΠατπ

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:={pq:p,q∈ℤ,q≠0}

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

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 ℤahlen

Natural Numbers 0
We define ℕ0:={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 ℕ1:={1,2,3,4,…}
Natural Numbers up to n
Let n∈ℕ1, 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}