# Maths

Set Theory

## Definition 1.1

An object in a set is called an element of a set or is said to belong to the set.If an object x is an element of a set A this is denoted by $x\in A$

If an object x is not a member of a set A this is denoted by $x\notin A$

## Definition 1.2

A set A is called a subset of set B if every element of the set A is an element of the Set B. If A is a subset of B this is is denoted by $A\subseteq B$

If set A is not a subset of set B this is denoted by $A⊈B$

## Definition 1.3

A set A is equal to a set B if $A\subseteq B$ and $B\subseteq A$

## Definition 1.4

The intersection of two sets A and B, denoted by $A\cap B$ is the set consisting of all elements contained in both A and B.

## Definition 1.5

The union of two sets A and B, denoted by $A\cup B$ is the set consisting of all elements contained in either A or B.

## Definition 1.6

The set difference, denoted by $B-A$ is the set of all elements in the set B that are not in the set A.

## Definition 1.7

The symmetric difference, denoted by $A△B$ is the set $\mathrm{\left(A}-\mathrm{B\right)}\cup \mathrm{\left(B}-\mathrm{A\right)}$
It is easily seen that $A△B$ = $\mathrm{\left(A}\cup \mathrm{B\right)-\left(A}\cap \mathrm{B\right)}$

## Definitions

The empty set denoted by $\varnothing$ or $\left\{\right\}$

## Definitions

The universe, or universe of discourse denoted by $U$

## Definition 1.8

Let A be a set. ${A}^{\text{'}}=U-A$ is the set of all elements not in A.

## Definition 1.9

The size or cardinality of a finite set A, denoted by $|A|$ , is the number of elements in the set.

An infinite set which can be listed so that there is a first element, second< element, third element etc. is called countably infinite.

If it cannot be listed it is said to be uncountable.

Two infinite sets have the same cardinality if there is a one-to-one correspondence between the two sets

We denote this by $|A|=|B|$

If there is a one-to-one correspondence between A and a subset of B we denote this by $|A|\le |B|$

If $|A|\le |B|$ but there is no one-to-one correspondence between A and B we denote this by $|A|<|B|$

## Definition 1.10

Let A and B be sets. The Cartesian product of A and B, denoted by $A×B$ is the set

For example let $A=\left\{a,b\right\}$ and $B=\left\{1,2,3\right\}$ , then $A×B=\left\{\left(a,1\right)\left(a,2\right)\left(a,3\right)\left(b,1\right)\left(b,2\right)\left(b,3\right)\right\}$

The Cartesian plane $R×R$ is the set of all ordered pairs of real numbers.

Note that for finite sets $|A×B|=|A|×|B|$

## Definition 1.11

The power set of set A, denoted by $P\left(A\right)$ is the set of all subsets of A.

For example the power set of $\left\{\mathrm{a,b,c}\right\}$ is
$\left\{\left\{a\right\},\left\{b\right\},\left\{\mathrm{c\right\},\left\{a,b\right\},\left\{a,c\right\}\left\{b,c\right\},\left\{a,b,c\right\},\varnothing \right\}}$

In the finite case it can be shown that $|P\left(A\right)|={2}^{|A|}$