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SETS

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                                             SETS 

     INTRODUCTION 

         We often speak some words in daily life which represent a collection of things, for example, a team of players, a bunch of flowers, a cluster of trees, a group of friends, a swarm of birds and so on. But in mathematics, we can use a single word "set" to denote all such type of words that show the collection of things such as, a set of players, a set of flowers, a set of trees, a set of friends and a set of birds, etc. Thus we can define a set as:
    "A collection of distinct and well-defined objects is called a set. "
 The objects of a set are called its members or elements.
    
    In 19th century George  Cantor was the first mathematician who gave the proper idea of sets that we are using now in various branches of mathematics.

    Well- defined sets 

       We-defined means a specific property of an object that enables it to be an element of a set or not. To make it clear consider the following examples of collections.
  
  (1)  The collection of good stories
  (2)  The collection of tasty foods
  (3)  The collection of favorite poems

      In the above examples, we can examine that words good, tasty and favorite are not well-defined because a food may be favorite of one person but may not be for another. Similarly, a story may be good in view of one person but may not be for another. So, these are not suitable examples of sets as these are not well defined.

   Distinct means the same objects should not appear more than once. For example, the set of letters of the word "small" is { s, m, a, l}. In this example, we can see that letter ' l ' has been written only once. If it is written twice then it is not a set.

     Set Notions 

    A set is represented by a capital letter A, B, C, . . . ,Z of English alphabets and its members or elements are written within brackets { } separated by conman, e. g.
 
                  Set of pet:  A={cow, horse, goat . . .}

Symbolically, we can write the members of the set A as,

      cow ∈ "A" is read as cow is an element of the set A
      goat ∈ "A" is read as goat is an element of  the set A and so on.

Now examine whether a tree is the element of the set A . No! a tree is not  an elements of the set A. So, we can write this statement symbolically as : 

 tree ∉ A is read as tree is not the element of the set A.

     The ∈is a Greek letter which is used is to tell that an object " is an element of " or "belongs to " or " is a member of " a set and symbol ∉ mean " does not belong to " or " is not the element of " the set.

Some important sets are given below:
  •  N = set of natural numbers 
  • W = set of whole numbers 
  •  E  = set of even numbers
  •  p  = set of prime numbers 
  •  O = set of odd numbers 

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