Math, asked by Madhuksaranjana, 1 month ago

What is meant by Exactly two of A or B or C (Set Chapter)​

Answers

Answered by atythh
0

Answer:

Step-by-step explanation:

n 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 v

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) Various 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

2020-21

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}.

2020-21

SETS 3

ANote In roster form, the order in which the elements are listed is immaterial.

Thus, the above set can also be represented as {1, 3, 7, 21, 2, 6, 14, 42}.

(b) The set of all vowels in the English alphabet is {a, e, i, o, u}.

(c) The set of odd natural numbers is represented by {1, 3, 5, . . .}. The dots

tell us that the list of odd numbers continue indefinitely.

ANote It may be noted that while writing the set in roster form an element is not

generally repeated, i.e., all the elements are taken as distinct. For example, the set

of letters forming the word ‘SCHOOL’ is { S, C, H, O, L} or {H, O, L, C, S}. Here,

the order of listing elements has no relevance.

Similar questions