Sets (NCERT) Notes | EduRev

: Sets (NCERT) Notes | EduRev

 Page 1


Chapter 1
SETS
Georg Cantor
(1845-1918)
 In these days of conflict between ancient and modern studies; there
must surely be something to be said for a study which did not
begin with Pythagoras and will not end with Einstein; but
is the oldest and the youngest.   — G.H. HARDY 
1.1  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 (1845-1918). He first
encountered sets while working on “problems on trigonometric
series”. In this Chapter, we discuss some basic definitions
and operations involving sets.
1.2  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:
(i) Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9
(ii) The rivers of India
(iii) The vowels in the English alphabet, namely, a, e, i, o, u
(iv) V arious kinds of triangles
(v) Prime factors of 210, namely, 2,3,5 and 7
(vi) The solution of the equation: x
2 
– 5x + 6 = 0, viz, 2 and 3.
We note that each of the above example is a well-defined collection of objects in
© NCERT
not to be republished
Page 2


Chapter 1
SETS
Georg Cantor
(1845-1918)
 In these days of conflict between ancient and modern studies; there
must surely be something to be said for a study which did not
begin with Pythagoras and will not end with Einstein; but
is the oldest and the youngest.   — G.H. HARDY 
1.1  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 (1845-1918). He first
encountered sets while working on “problems on trigonometric
series”. In this Chapter, we discuss some basic definitions
and operations involving sets.
1.2  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:
(i) Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9
(ii) The rivers of India
(iii) The vowels in the English alphabet, namely, a, e, i, o, u
(iv) V arious kinds of triangles
(v) Prime factors of 210, namely, 2,3,5 and 7
(vi) The solution of the equation: x
2 
– 5x + 6 = 0, viz, 2 and 3.
We note that each of the above example is a well-defined collection of objects in
© NCERT
not to be republished
2       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 real numbers.
The symbols for the special sets given above will be referred to throughout
this text.
Again the collection of five most renowned mathematicians of the world is not
well-defined, because the criterion for determining a mathematician as most renowned
may vary from person to person. Thus, it is not a well-defined collection.
We shall say that a set is a well-defined collection of objects.
The following points may be noted :
(i) Objects, elements and members of a set are synonymous  terms.
(ii) Sets are usually denoted by capital letters A, B, C, X, Y , Z, etc.
(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.
If a is an element of a set A, we say that “ a belongs to A” the Greek symbol ?
(epsilon) is used to denote the phrase ‘belongs to’. Thus, we write a ? A. If ‘b’ is not
an element of a set A, we write b ? A and read  “b does not belong to A”.
Thus, in the set V of vowels in the English alphabet, a ? V but b ? V . In the set
P of prime factors of 30, 3 ? P but 15 ? P.
There are two methods of representing a set :
(i) Roster or tabular form
(ii) Set-builder form.
(i) In roster form, all the elements of a set are listed, the elements are being separated
by commas and are enclosed within braces {   }. For example, the set of all even
positive integers less than 7 is described in roster form as {2, 4, 6}. Some more
examples of representing a set in roster form are given below :
(a) The set of all natural numbers which divide 42 is {1, 2, 3, 6, 7, 14, 21, 42}.
© NCERT
not to be republished
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