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 xA

If an object x is not a member of a set A this is denoted by xA

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 AB

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

Definition 1.3

A set A is equal to a set B if AB and BA

Definition 1.4

The intersection of two sets A and B, denoted by AB 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 AB 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 AB is the set (A-B)(B-A)
It is easily seen that AB = (AB) - (AB)

Definitions

The empty set denoted by or {}

Definitions

The universe, or universe of discourse denoted by U

Definition 1.8

Let A be a set. A'=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||B|

If |A||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 {(a,b):aA and  bB}

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

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(A) is the set of all subsets of A.

For example the power set of {a,b,c} is
{{a},{b},{c},{a,b},{a,c}{b,c},{a,b,c},∅}

In the finite case it can be shown that |P(A)|=2|A|