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 \( \mathbb{N}_{0} := \left \lbrace 0 , 1 , 2 , 3 , 4 , \ldots \right \rbrace \)
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 \( P \left ( X \right ) := \left \lbrace S : S \subseteq X \right \rbrace \)