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Basic Concepts of Set Theory
The modern approach to probability based on axiomatically defining probability as
function of a set. A background on the set theory is essential for understanding
probability. Some of the basic concepts of set theory are introduced here.
Set: A set is a well defined collection of objects. These objects are called elements or
members of the set. Usually uppercase letters are used to denote sets.
 Example 1
A  a, e, i, o, u is a set and a, e, i, o and u are its elements.

The elements of a set are enumerated within a pair of curly brackets as
shown in this example.

In stead of listing all the elements, we can represent the set in the setbuilder notation specifying some common properties satisfied by the
elements. Thus the set B  {x | 0  x  9 and x is an even integer}
represents the set {0,2,4,6,8}. We read ‘ x | ’ as ‘all x such that’.
Particularly, if a set is infinite having infinite number of elements or
listing all the elements of the set is cumbersome, such a
representation is useful.
If an element x is a member of the set A, we write x  A . If x is not a
member of A, we write x  A . In the above example, a  A and d  A .


The null set of empty set is the set that does not contain any element. A
null set is denoted by  .
Some commonly used sets and their standard notations:
1.
2.
3.
4.
5.
6.
Set of natural numbers N
Set of integers
Set of rational numbers Q
Set of real numbers
Set of complex numbers C
Open interval on the real line
 a, b  x | a  x  b, a, b  
7. Closed interval on the real line
a, b  x | a  x  b, a, b  
8, Semi open intervals
( a , b ]   x | a  x  b, a , b  
[a, b)   x | a  x  b, a, b 

Subset and Superset
A set A is called a subset of B ( or B is called the superset of A), denoted by
A  B or B  A. , if all the elements of A are also elements of B. Thus
A  B if and only if, x  A  x  B
If A is a subset of B and there is at least one element in B which is not an element of A,
then A is called a proper subset of B. We write A  B or B  A.
Example: Let A  1, 2,3,5 , B  1, 2,3, 4,5,6,7 and C  2, 4,6,8.
Then, A  B, C  A and B  C , etc.




A set A is a subset of itself A  A.
A  B and B  C implies that A  C.
The null set  is a subset of every set.
If the set A is finite with n number of elements, then A has 2n subsets.
For example, the set of binary digits B  {0,1} has 22  4 subsets. These
are:  ,{0},{1} and B.
Universal set
We always consider all sets for the problem under consideration to be a subset of a
(large) set called the universal set. For the binary digital communication problem, the set
B  {0,1} may be considered as the universal set. We shall denote the universal set by the
symbol S.
Example2
In discussion involving English letters, the alphabet of the English language may be
considered as the universal set.
Equality of two sets: Two sets A and B are equal if and only if they have the same
elements. Thus,
 A = B if and only if A  B and B  A . In other words,
A  B if and only if, x  A  x  B and , x  B  x  A
We take above definition of the equality of two sets to establish identities involving set
theoretic operation.
Set operations
We can combine events by set operation to get other events. Following set
operations are useful:
The union of two sets A and B, denoted by A  B is defined as the set of elements
that are either in A or in B or both. In set builder notation, we write
A  B  x | x  A or x  B
The intersection of two sets A and B, denoted by A  B , is defined as the set of
elements that are common to both A and B. We can write,
A  B  x | x  A and x  B
The difference of two sets A and B, denoted by A \ B, is the set of those elements
of A which do not belong to B. Thus,
A \ B  x | x  A and x  B
Similarly,
B \ A  x | x  B and x  A
The complement of a set A, denoted by AC , is defined as the set of all elements
which are not in A.
Ac  x | x  A 
Clearly, Ac  S \ A
Example:
Let S  {1, 2,3, 4,5,6,7,8,9}, A  1, 2,3,5 , B  2,3, 4,5 and C  6,8.
Then A  B  {1, 2,3, 4,5}, A  B  {2,3,5}, B  C   , A \ B  {1}, Ac  {4,6,7,8,9}, etc.
 Two sets A and B are called disjoint if A  B   . In the above example
B and C are disjoint sets.
The sets and set operations can be illustrated by means of the Venn diagrams. A
rectangle is used to represent the universal set and a circle is used to represent any set in
it. Following are a few illustrations of Venn diagrams:
Properties of set operations
1. Identity properties
A  A  A,
A  A  A,
A  S  S,
A  S  A,
and
A    A,
A   
2. Commutitive properties
A  B  B  A,
A B  B  A
3. Associative properties
A   B  C    A  B  C
A   B  C    A  B  C
4. Distributive properties
A   B  C    A  B   A  C 
A   B  C    A  B   A  C 
 Distributive properties 
5. Complementary properties
A  AC  S
A  AC  
6. De Morgan’s laws
C
 A  B   AC  BC
 A  B
C
 AC  BC
.
These properties can be proved easily and verified using the Venn diagram. They can be
used to drive useful results. For example,
Putting C  B c in
A   B  C    A  B   A  C 
we get
A   B  Bc    A  B    A  Bc 
 A  S   A  B    A  Bc 
 A   A  B    A  Bc 
Cartesian Product of two Sets:
Consider two sets A and B, not necessarily from the same universal set. Then the set of
all ordered pairs ( x, y), where x  A and y  B, called the Cartesian product of A and B
and is denoted by A  B. Thus
A  B  {( x, y) | x  A, y  B}
Example: Suppose A  {1, 2,3} and B  {2, 4}.
Then A  B  {(1, 2), (1,4), (2,2), (2,4), (3,2), (3,4)}
A  A  {(1,1), (1, 2), (1,3), (2,1), (2, 2), (2,3), (3,1), (3, 2), (3,3)}
Remark


-
A  A is denoted by A2 .
Some important Cartesian products
2
  .
2
 
We can extend the set theoretic operations to arbitrary number of sets. Particularly, if we
have a sequence of sets A1 , A2 , A3 ,..............., An ,....... we can define

i 1

Ai   x | x  Ai for some i  N 
Ai   x | x  Ai , i  N 
i 1
Thus we can define the union and the intersection of countably infinite number of sets.
Power set
The set of all the subsets of a set S is called the power set of S and denoted by Ps . Thus
Ps   X | X  S
Note the following

A collection of sets is called a class of sets. Each element of a class is a
set. Thus Ps is a class.


If S contains n elements, Ps contains 2 n elements.
If S is finite, then Ps is closed under the set theoretic operation of unions,
intersection and complementation. Thus for all X , Y  Ps ,
X C  PS
X  Y  PS
X  Y  PS

If S is an infinite set, Ps may not be closed under countably infinite unions
or countably infinite intersections. Such situations lead to serious
mathematical problems.
This necessitates defining a subset of Ps which is closed under these countably infinite
set theoretic operations. Such a family of sub sets of Ps is called sigma-algebra and
defined below.
  Algebra of Sets or   Field of Sets
Let S be a set. A sigma field F is a subset of the power set PS with the following
properties:
(1).S  F .
(2). If A F,
AC  F .
(3). If for 1  n  , An  F then

i 1
An  F .
Thus F is closed under complementation and countable union operations.
Remark:
 From (1) and (2) it follows that   S c  F.
 {S , } is the smallest sigma-algebra formed from the subsets of S .
 If S is a finite set, then PS is a sigma-algebra.
F is closed under countable intersection operations also. Suppose

for 1  n  , An  F . Then Anc  F and therefore,

n 1

An  ( Anc )c  F .
n
Example: Suppose S  {0,1, 2}. Then the power set
PS  {{0},{1},{2},{0,1},{1, 2},{0, 2}, S , } is a sigma algebra.
Example: The collection {S , } is the smallest sigma algebra containing S . For a finite
set S , the power set Ps is the largest sigma algebra containing S .
Example: Consider the class C of all open intervals of the form (a, b) on the real line.
Here S  R and C  {(a, b) | a  , b  }.
Consider the sequence of subsets in C given by
1
1
An  (1  ,1  )
n
n
It can be shown that

n 1
(1 
1
1
,1  )  [1,1]
n
n
Hence the collection of open intervals of the real line is not a sigma-algebra.
Remark
The minimum sigma field containing C is called the Borel field and denoted by . The
same Borel field is obtained if C is considered as the class of all closed intervals of the
form [a, b] or the semi-open intervals of the form (a, b] or [a, b). It can be shown that
 contains all the open intervals, closed intervals and semi-open intervals.