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 \( \left \lbrace 1 , 2 , 3 , 41 \right \rbrace \)

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 \( [n] := \{1, 2, \ldots, n\} \)