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Sets

INTRODUCTION OF SETS:

  The concept of sets is being used in almost every branch of mathematics. Sets are used to defind the concepts of relations and functions. The study of geometry sequences, probability etc. requires the knowledge of sets.

  The theory of sets was developed by German mathematician George Cantor (1845 – 1918).  He first encountered sets while working on “ Problems on trigonometric series “.

 

Now let us discuss about Sets :

 

In our daily life we often speak of collection of objects of a particular kind, such as crowd of people, bunch of flowers cricket team etc. Similarly in Mathematics also we come across collections such as Natural numbers, Odd numbers,  Even numbers, Prime numbers etc.

Specially we examine collections like

i)                   Even natural numbers less than 10 i.e., 2, 4, 6, 8

ii)                 Vowels of English alphabets  a, e, I, o, u.

iii)               Solutions of x+7=0 i.e., -7

We observe that all of these are well defined collection of objects.

Let us discuss few more examples which are used particularly in mathematics.

i)                   N : the set of Natural numbers

ii)                 W : The set of whole numbers

iii)               Z : The set of integers

iv)               Z+ : The set of positive integers

v)                 Q : the set of rational numbers

vi)               Q+ : the set of positive rational numbers.

vii)             R : The set of real numbers

Now let us discuss about Well –defined .

Collection of five renowned mathematicians of the world.

This is not well defined , because for determining a mathematician as most renowned may vary from person to person.

We defined a Set as a “well defined collection of objects “.

Objects of a set are called elements of a set.

Sets are usually denoted by capital letters A, B, C, X, Y, Z……

Elements are represented by small letters a, b, c, x, y, z ……

Ex:    Let A  is the set of  elements 1,2, 3, 4

     1 is the element of A, we say that 1 belongs to A . 1 ε A

  5 is not an element of A we say as 5 does not belongs to A.

Now Let us discuss methods of Representing a set.



 


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