Modern Theory of probability is based on set theory,so before reading probability we need to understand the concept of sets.

Sets (Ω) - space or universal sets.

"A Set is a collection of well-defined distinct objects".


Example 1:-The collection of positive number less than 10 is a set,so for any given number we can say number belongs to set or not. But the collection of good students in your class is not set as in this case no defined rule for student to be good and bad.

Example 2:-Collection of first five month of year is set but collection of rich man in city is not set.

Elements of sets-

The different objects that form a set are called elements of set.These elements can be in any order are not repeated.Elements are denoted by small letters.
If x is element of set A then mathematically x⊂A (x belongs to A).And if x is not element of A then mathematically x⊄A.

Example - Collection of vowels in English alpabets. V=[a,e,i,o,u].

Some important sets used in mathematics are:-

N - The sets of all natural number = {1,2,3,4,...}
Z - The sets of all integer = {...,-3,-2,-1,0,1,2,3,...}
Z+- The sets of all positive numbers.
Q - The sets of all rational numbers.
R - The sets of all real numbers.
W - The sets of all whole numbers.
C - The sets of all complex numbers.

Properties of sets-

1. Order does not matters {2,3,4}={4,2,3}.
2. If elements are repeated in set then set will be same with unique elements like {1,1,2,3,2,4,5,5}={1,2,3,4,5}.

Representation of sets-

1. Statement form method
2. Roster or Tabular form method
3. Rule or Set builder form method

Statement form method:
A set of odd numbers less than 7 will be represent in stateform as {odd numbers less than 7}.

Roster or tabular form method:

Set builder form method:
Suppose P is set of counting numbers greater than 12 then set P in set builder can be witten as:
P = {x:x is counting number and greater than 12} OR P = {x|x is counting number and greater than 12}

Types of sets-

Empty sets or Null sets(∅): ∅={}
Example:-The set of whole number less than 0 is empty set because there is not whole number less than 0.
Example:- N = {x:x⊂N,3<x<4}.It is empty set there is no integer number which can be greather than 3 and less than 4.

Singleton sets(∅):
A set which contain only one element is called singleton set.
Example:- X={x:x is neither prime number or composite} = {1}