is the oldest and the youngest. — G.H. HARDY v . Introduction The concept of set serves as a fundamental part of the present day mathematics.
Today this concept is being used in almost every branch of mathematics. Sets are used to define 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 Georg Cantor ( - ). He first encountered sets while working on “problems on trigonometric series”. In this Chapter, we discuss some basic definitions and operations involving sets. .
Sets and their Representations In everyday life, we often speak of collections of objects of a particular kind, such as, a pack of cards, a crowd of people, a cricket team, etc. In mathematics also, we come across collections, for example, of natural numbers, points, prime numbers, etc. More specially, we examine the following collections: Odd natural numbers less than , i.e., , , , , The rivers of India (iii) The vowels in the English alphabet, namely, a, e, i, o, u (iv) Various kinds of triangles (v) Prime factors of , namely, , , and (vi) The solution of the equation: x – x + = , viz, and . We note that each of the above example is a well-defined collection of objects in MATHEMATICS the sense that we can definitely decide whether a given particular object belongs to a given collection or not.
For example, we can say that the river Nile does not belong to the collection of rivers of India. On the other hand, the river Ganga does belong to this colleciton. We give below a few more examples of sets used particularly in mathematics, viz. N : the set of all natural numbers Z : the set of all integers Q : the set of all rational numbers R : the set of real numbers Z + : the set of positive integers Q + : the set of positive rational numbers, and R + : the set of positive