# Sets

An Overview of Sets

A

**set**is a collection of**elements**. When defining a set, we often use curly braces$S = \{\dots\}$

. The number of elements in a set is denoted $\left| S \right|$

or $n(S)$

. We can say an element $a$

is a **member**of the set$S$

by saying $a \in S$

. Note that a set has **no repeating elements**.Common sets have their own symbols:

$\N$

- the set of **natural numbers**(there is no agreement on whether this contains$0$

or not)

$\mathbb{Z}$

- the set of **integers**

$\mathbb{Q}$

- the set of **rational numbers**

$\mathbb{R}$

- the set of **real numbers**

$\mathbb{C}$

- the set of **complex numbers**This notation allows us to easily define sets. For example, if we wanted to define a set with all multiple of 3 up to 30, we could do the following:

$S = \{ x : x = 3k, k \in \N, x \leq 30 \}$

Assuming we have two sets

$A$

and $B$

, we have a way to interact with them.We say

$A = B$

if both sets have the exact same elements.The set

$U$

contains all elements of interest.The union of two sets

$A \cup B$

is a combination set of all elements in $A$

or $B$

.The intersection of two sets

$A \cap B$

is a set of all elements that appear in **both**$A$

and $B$

.The set-difference of

$A$

and $B$

is the set of all elements in $A$

not in $B$

.$A \setminus B = \{ a \in A : a \not \in B\}$

The complement of

$A$

, denoted $A^\prime$

, is all elements of $U$

not in $A$

. You can think of it as $U \setminus A$

.If all elements of

$A$

are also elements of $B$

, then $A$

is a subset of $B$

and this is denoted $A \subseteq B$

. Note that means it **is possible that**$A = B$

. To discount this possibility, we say $A$

is a **proper subset**of$B$

, denoted $A \subset B$

.