BASIC DEFINITIONS

1)     Set: A well-defined collection of distinct objects is called a set.

2)     Notation of Sets: Capital letters are usually used to denote or represent a set.

3)     Representation of Sets: There are two methods of representing a set. 
     (i) Roster Method 
     (ii) Set builder form.

4)     Finite and Infinite Sets: A set is finite if it contains a specific number of elements. Otherwise, a set is an infinite set.

5)     Null Set or Empty Set or Void Set: A set with no elements is an empty set .Denoted by { } or  Ø

6)     Equivalent Sets: Two finite sets A and B are said to be equivalent sets if cardinality of both sets are equal
     i.e. n (A) = n (B).

7)     Equal Sets: Two sets A and B are said to be equal if and only if they contain the same elements 
    i.e. if every element of A is in B and every element of B is in A. We denote the equality by A = B.

8)    Cardinality of a Set A: The number of elements in a finite set A, is the cardinality of A and is denoted by n(A).

9)     Universal Set: In any application of the theory of sets, the members of all sets under consideration usually belong to some fixed large set called the universal set.

10)  Subsets: If A and B are sets such that each element of A is an element of B, then we say that A is a subset of B and write A Í B.

11)  Power Set: The family of all subsets of any set S is called the power set of S. We denote the power set of S by P (S).

Set of number.

 i)  set of natural numbers , N={ 0,1,2,3,4, … }

ii) set  of positive integers, P={ 1, 2, 3, 4, … }

iii) set of all integers, Z= { …., -3, -2,-1,0,1,2,3,… }

iv) set of all real numbers, R= { …, -3, -2.5, -1/2, 0, 1, 3/2, … }

APPLICATION OF SET

       Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

       One of the main applications of naive set theory is constructing relations. A relation from a domainA to a codomainB is a subset of the Cartesian product A × B. Given this concept, we are quick to see that the set F of all ordered pairs (x, x²), where x is real, is quite familiar. It has a domain set R and a codomain set that is also R, because the set of all squares is subset of the set of all reals. If placed in functional notation, this relation becomes f(x) = x². The reason these two are equivalent is for any given value, y that the function is defined for, its corresponding ordered pair, (y, y²) is a member of the set F.

       Application of Sets is the important chapter in the study of sets. The set theory is the basic frame work of all branches of mathematics. Union and Intersection are the basic set operations. Using this operations we have to prove the laws of sets. They are Distributive law and Associative law and Demorgan’s law.