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
If an object x is not a member of a set A this is denoted by
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
If set A is not a subset of set B this is denoted by
A set A is equal to a set B if and
The intersection of two sets A and B, denoted by is the set consisting of all elements contained in both A and B.
The union of two sets A and B, denoted by is the set consisting of all elements contained in either A or B.
The set difference, denoted by is the set of all elements in the set B that are not in the set A.
The symmetric difference, denoted by
is the set
It is easily seen that =
The empty set denoted by or
The universe, or universe of discourse denoted by
Let A be a set. is the set of all elements not in A.
The size or cardinality of a finite set A, denoted by , 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
If there is a one-to-one correspondence between A and a subset of B we denote this by
If but there is no one-to-one correspondence between A and B we denote this by
Let A and B be sets. The Cartesian product of A and B, denoted by is the set
For example let and , then
The Cartesian plane is the set of all ordered pairs of real numbers.
Note that for finite sets
The power set of set A, denoted by is the set of all subsets of A.
For example the power set of
In the finite case it can be shown that