Class 11 Maths NCERT Solutions of the Chapter 1-Sets- PDF

Download PDF of NCERT solutions of class 11 maths chapter 1-Sets. All the NCERT solutions are solved by an expert teacher of maths. NCERT solutions of class 11 maths chapter 1 – Sets are very important for the students of class 11. These NCERT Solutions of the chapter 1-sets provide complete information of all kinds of operations on sets.All unsolved questions of class 11 maths NCERT chapter 1 Set are solved in the NCERT Solutions of Class 11 Maths of ch.1-Sets(PDF).

A set in mathematics is a collection of well defined and distinct objects. Sets are one of the most fundamental concepts in mathematics.  Set theory is used everywhere in mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a higher maths.

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Class 11 Maths NCERT Solutions of the Chapter 1-Sets

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In everyday life, we often speak of collections of objects of a particular kind, such as, Tea set, Dining set, 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, rational numbers, irrational numbers, etc. More specially, we examine the following collections:

(i)Odd natural numbers less than 20, i.e., 1, 3, 5, 7, 9 ,11,13,15,17,19

(ii) The rivers of India originated from Himalaya

(iii) The vowels in the English alphabet, namely, a, e, i, o, u

(iv) Various kinds of triangles

(v) Prime factors of 30, namely, 2,3,5

(vi) The solution of the equation: x² – 5x + 6 = 0, viz, 2 and 3. We note that each of the above examples is a well-defined collection of objects in 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 in India. On the other hand, the river Ganga does belong to this collection.

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,

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 scientists of the world is not well-defined, because the criterion for determining a scientist as most renowned may vary from person to person. Thus, it is not a well-defined collection.

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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.

(i)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}.

(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 continues indefinitely.

(ii) In the set-builder form, all the elements of a set possess a single common property that is not possessed by any element outside the set. For example, in the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possesses this property. Denoting this set by V, we write

V = {x : x is a vowel in English alphabet}

It may be observed that we describe the elements of the set by using a symbol x (any other symbol like the letters y, z, etc. could be used) which is followed by a colon “ : ”. After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces. The above description of the set V is read as “the set of all x such that x is a vowel of the English alphabet”. In this description the braces stand for “the set of all”, the colon stands for “such that”. For example, the set

A = {x : x is a natural number and 3 < x < 10} is read as “the set of all x such that x is a natural number and x lies between 3 and 10. Hence, the numbers 4, 5, 6, 7, 8, and 9 are the elements of the set A.

If we denote the sets described in (a), (b) and (c) above in roster form by A, B, C, respectively, then A, B, C can also be represented in set-builder form as follows:

A= {x : x is a natural number which divides 42}

B= {y: y is a vowel in the English alphabet}

C= {z: z is an odd natural number} Example 1 Write the solution set of the equation

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