NCERT Solutions for Class 11 Maths Chapter 1 Sets

Collecting and Segregating things is a fundamental task that we all use in our day-to-day lives to organize things. In Mathematics, too, we use the process of collecting and segregating elements in a well-defined manner, and it is known as Sets. Sets are well-defined collections of objects, and they can be classified into different types based on their elements or characteristics, including finite, infinite, empty, singleton, equal, and universal sets. It also introduces Intervals as subsets of the real numbers (R), the Universal Set, Venn Diagrams, and operations on Sets, such as Union, Intersection, Difference, and Complement.

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This Story also Contains

1. Sets Class 11 Questions And Answers PDF Free Download
2. Set Class 11 Solutions – Important Notes
3. Set Class 11 NCERT Solutions (Exercise)
4. Sets Class 11 Maths Solutions - Chapter Wise
5. Sets Class 11 Solutions - Subject Wise
6. Importance of solving NCERT questions for Chapter 1 Sets of Class 11
7. NCERT Books and NCERT Syllabus

This article on NCERT Solutions for Class 11 Maths Chapter 1 Sets provides clear and step-by-step solutions for exercise problems in the NCERT Class 11 Maths Book. These solutions for class 11 maths chapter 1 Sets are designed by Subject Matter Experts according to the latest CBSE syllabus, ensuring that students grasp the concepts effectively. Students can also refer to the notes that will help them to understand the concept in a better manner. Sets Notes. Also, after practicing all the questions, students can try the advanced questions of NCERT Exemplar and can refer to the solutions created by our experts in NCERT Exemplar Solutions For Sets. NCERT solutions for other subjects and classes can be downloaded from NCERT Solutions for Class 11.

Sets Class 11 Questions And Answers PDF Free Download

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Set:

A set is a well-defined collection of objects. It can be represented in 2 forms, i.e. Set Builder and Roaster forms.

Examples of sets:

$\mathbf{N}$: the set of all natural numbers

$\mathbf{Z}$: the set of all integers

$\mathbf{Q}$: the set of all rational numbers

$\mathbf{R}$: the set of real numbers

${\mathbf{Z}}^{+}$: the set of positive integers

${\mathbf{Q}}^{+}$: the set of positive rational numbers, and

${\mathbf{R}}^{+}$: the set of positive real numbers.

Type of Sets:

Empty Set or Null Set: It has no element present in it. Example: $A=\{\}$ is a null set.

Finite Set: It has a limited number of elements. Example: $A=\{1,2,3,4,6,8,9\}$

Infinite Set: It has an infinite number of elements. Example: $A=\{x:x$ is the set of all Complex numbers $\}$

Equal Set: Two sets which have the same members. Example: $A=\{1,2,8\}$ and $B=\{2,8,1\}$ : Set $A=\mathrm{Set}B$

Subsets: $A$ set ' $A$ ' is said to be a subset of $B$ if each element of $A$ is also an element of B. Example: $A=\{1,2,4\},B=\{1,2,3,4,5,6\}$, then $A\subseteq B$

Universal Set: A set which consists of all elements of other sets present in a Venn diagram. Example: $A=\{1,3\},B=\{2,3\}$, The universal set here will be, $U=\{1,2,3\}$

Intervals:

An interval is a set of real numbers that includes all numbers between two specified endpoints, between one endpoint and infinity, or between negative infinity and one endpoint.

Types Of Intervals:

Open Interval: An interval that does not include its endpoints. It's represented using parentheses () in interval notation.

Closed Interval: An interval that includes its endpoints. It's represented using square brackets [] in interval notation.

Half-Open/Half-Closed Interval: An interval that includes one endpoint but not the other. It's represented using a combination of parentheses and square brackets.

Important Formulae

Union of Sets $(A\cup B):A\cup B$

Intersection of Sets $(A\cap B):A\cap B$

Complement of a Set $(A^{\prime }):A^{\prime }$

De Morgan’s Theorem:

De Morgan’s Theorems state the relationship between union and intersection in set theory and Boolean algebra. The two theorems are:

First Theorem:

$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$

Second Theorem:

$(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$

Set Cardinality with Intersection:

$n(A\cup B)=n(A)+n(B)$

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$

Other Set Formulas:

$A-A=\varphi$

$B-A=B\cap A^{\prime }$

$B-A=B-(A\cap B)$

$(A-B)=A$ if $A\cap B=\varphi$

$(A-B)\cap C=(A\cap C)-(B\cap C)$

$A\mathrm{△}B=(A-B)\cup (B-A)$

$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(B\cap C)-n(A\cap B)-n(A\cap C)+n(A\cap B\cap C)$

$n(A-B)=n(A\cup B)-n(B)$

$n(A-B)=n(A)-n(A\cap B)$

$n(A^{\prime })=n(U)-n(A)$

$n(U)=n(A)+n(B)-n(A\cap B)$

$n((A\cup B{)}^{\prime })=n(U)+n(A\cap B)-n(A)-n(B)$

Set Class 11 NCERT Solutions (Exercise)

NCERT sets class 11 questions and answers - Exercise: 1.1(Page no: 4, Total Question)

Question 1 (i) Which of the following are sets? justify your answer

The collection of all the months of a year beginning with the letter J.

Answer:

The months starting with the letter J are:

January

June

July

Hence, this is a collection of well-defined objects, so it is a set.

Question 1 (ii) Which of the following are sets? justify your answer

The collection of the ten most talented writers of India.

Answer:

The ten most talented writers may be different depending on different criteria for determining talented writers.

Hence, this is not well-defined, so it cannot be a set.

Question 1 (iii) Which of the following are sets? justify your answer

A team of eleven best-cricket batsmen of the world.

Answer:

The eleven most talented cricketers may be different depending on the criteria for determining the talent of a player.

Hence, this is not well defined, so it is not a set.

Question 1 (iv): Which of the following are sets? justify your answer

The collection of all boys in your class.

Answer:

The collection of boys in a class is well-defined and known.

A group of well-defined objects is a set.

Hence, it is a set.

Question 1 (v) Which of the following are sets? justify your answer

The collection of all natural numbers is less than 100.

Answer:

Natural numbers less than 100 have a defined and known collection of numbers.

that is S= {1,2,3............99}

Hence, it is a set.

Question 1 (vi) Which of the following are sets? Justify your answer

A collection of novels written by the writer Munshi Prem Chand.

Answer:

The collection of novels written by Munshi Prem Chand is well-defined and known.

Hence, it is a set.

Question 1 (vii) Which of the following are sets? Justify your answer

The collection of all even integers.

Answer:

The collection of even integers is well-defined because we can get even integers till infinity. that is

Hence, it is a set.

Question 1 (viii) Which of the following are sets? Justify your answer

The collection of questions in this Chapter.

Answer:

The collection of questions in a chapter is well-defined and known.

Hence, it is a set.

Question 1 (ix) Which of the following are sets? Justify your answer

A collection of the most dangerous animals in the world.
Answer:

A Collection of the most dangerous animals is not well defined because the criteria for defining the dangerousness of any animal can vary.

Hence, it is not a set.

Question 2 $LetA=\{1,2,3,4,5,6\}.Inserttheappropriatesymbol\in or\notin intheblankspace:$

(i) 5_____A

(ii) 8_____A

(iii) 0_____A

(iv) 4_____A

(v) 2_____A

(vi) 10____A

Answer:

A = $\{1,2,3,4,5,6\}$, the elements which lie in this set belong to this set, and others do not belong.

(i) 5 $\in$ A

(ii) 8 $\notin$ A

(iii) 0 $\notin$ A

(iv) 4 $\in$ A

(v) 2 $\in$ A

(vi) 10 $\notin$ A

Question 3(i) Write the following sets in roster form

$\text{(i)}A=\{x:x\text{ is an integer and }-3\le x<7\}$

Answer:

Elements of this set are: -3, -1, 0, 1, 2, 1,0,1,2,3,4,5,6.

Hence, this can be written as:

A = $\{-3,-2,-1,0,1,2,3,4,5,6\}$

Question 3 (ii) Write the following sets in roster form

$\left(ii\right)B=\{x:xisanaturalnumberlessthan6\}$

Answer:

Natural numbers less than 6 are, and 1,2,3,4,5.

This can be written as:

$B=\{1,2,3,4,5\}$

Question 3(iii) Write the following sets in roster form

C = {x: x is a two-digit natural number such that the sum of its digits is 8}

Answer:

The two-digit numbers having a sum equal to 8 are, and 17,26,35,44,53,62,71,80.

This can be written as:

$C=\{17,26,35,44,53,62,71,80\}$

Question 3 (iv) Write the following sets in roster form

D = {x: x is a prime number which is a divisor of 60}

Answer:

$60=2\times 2\times 3\times 5$

Prime numbers which are divisors of 60 are 2, 3, and 5.

This can be written as:

$D=\{2,3,5\}$

Question 3(v) Write the following sets in roster form

E = The set of all letters in the word TRIGONOMETRY.

Answer:

Letters of the word TRIGONOMETRY are: T, R, I, G, N, O, M, E, Y.

This can be written as:

E = {T, R, I, G, N, O, M, E, Y}

Question 3 (vi) Write the following sets in roster form

F = The set of all letters in the word BETTER.

Answer:

The set of letters of the word BETTER are: {B, E, T, R}

This can be written as:

F = {B, E, T, R}

Question 4 (i) Write the following sets in the set builder form

{3, 6, 9, 12}

Answer:

A = {3,6,9,12}

This can be written as: $\{3,6,9,12\}=\{x:x=3n,n\in Nand1\le n\le 4\}$

Question 4(ii) Write the following sets in the set builder form

{2,4,8,16,32}

Answer:

$2=2^1$

$4=2^2$

$8=2^3$

$16=2^4$

$32=2^5$

$\{2,4,8,16,32\}$ can be written as $\{x:x=2^n,n\in Nand1\le n\le 5\}.$

Question 4 (iii) Write the following sets in the set builder form:

{5, 25, 125, 625}

Answer:

$5=5^1$

$25=5^2$

$125=5^3$

$625=5^4$

$\{5,25,125,625\}$ can be written as $\{x:x=5^n,n\in Nand1\le n\le 4\}$

Question 4 (iv) Write the following sets in the set builder form:

{2, 4, 6, . . .}

Answer:

This is a set of all even natural numbers.

{2,4,6....} can be written as {x: x is an even natural number}

Question 4 (v) Write the following sets in the set builder form :

{1,4,9, . . .,100}

Answer:

$1=1^2$

$4=2^2$

$9=3^2$

.

.

.

$100={10}^2$

$\{1,4,9.....100\}$ can be written as $\{x:x=n^2,n\in Nand1\le n\le 10\}$

Question 5(i) List all the elements of the following sets:

A = {x: x is an odd natural number}.

Answer:

A = { x : x is an odd natural number } = {1,3,5,7,9,11,13.............}

Question 5 (ii) List all the elements of the following sets:

B= { x : x is an integer, $\frac{-1}{2}<x<\frac{9}{2}$ }

Answer:

Integers between $-\frac{1}{2}<x<\frac{9}{2}$ are 0,1,2,3,4.

Hence, B = {0,1,2,3,4}

Question 5 (iii) List all the elements of the following sets:

C = {x : x is an integer, $x^2\le 4$ }

Answer:

${(-2)}^2=4$

${(-1)}^2=1$

${\left(0\right)}^2=0$

${\left(1\right)}^2=1$

${\left(2\right)}^2=4$

Integers whose square is less than or equal to 4 are: -2,-1,0,1,2.

Hence, it can be written as c = {-2, -1,0,1,2}.

Question 5 (iv) List all the elements of the following sets:

D = {x: x is a letter in the word “LOYAL”}

Answer:

LOYAL has the letters L, O, Y, A.

D = {x: x is a letter in the word “LOYAL”} can be written as {L, O, Y, A}.

Question 5 (v): List all the elements of the following sets:

E = {x: x is a month of a year not having 31 days}

Answer:

The months not having 31 days are:

February

April

June

September

November

It can be written as {February, April, June, September, November}

Question 5 (vi) List all the elements of the following sets :

F = {x: x is a consonant in the English alphabet which precedes k }.

Answer:

The consonants in English which precede K are: B, C, D, F, G, H, J

Hence, F = {B, C, D, F, G, H, J}.

Question 6 Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form:

(i) {1, 2, 3, 6} (a) {x: x is a prime number and a divisor of 6}

(ii) {2, 3} (b) {x: x is an odd natural number less than 10}

(iii) {M, A, T, H, E, I, C, S} (c) {x: x is a natural number and a divisor of 6}

(iv) {1, 3, 5, 7, 9} (d) {x: x is a letter of the word MATHEMATICS}.

Answer:

(i) 1,2,3,6, all are natural numbers and also divisors of 6.

Hence, (i) matches with (c)

(ii) 2,3 are prime numbers and are divisors of 6.

Hence, (ii) matches with (a).

(iii) M, A, T, H, E, I, C, S are letters of the word "MATHEMATICS".

Hence, (iii) matches with (d).

(iv) 1,3,5,7,9 are odd natural numbers less than 10.

Hence, (iv) matches with (b).

NCERT class 11 maths chapter 1 question answer sets-Exercise: 1.2 (Page no: 8, Total Question 6)

Question 1(i) Which of the following are examples of the null set :

Set of odd natural numbers divisible by 2

Answer:

No odd number is divisible by 2.

Hence, this is a null set.

Question 1 Question 1 estion 1 (ii) Which of the following are examples of the null set :

Set of even prime numbers.

Answer:

Even prime number = 2.

Hence, it is not a null set.

Question 1(iii) Which of the following are examples of the null set:

{ x : x is a natural numbers, $x<5$ and $x>7$ }

Answer:

No number exists which is less than 5 and more than 7.

Hence, this is a null set.

Question 1(iv) Which of the following are examples of the null set :

{y: y is a point common to any two parallel lines}

Answer:

Parallel lines do not intersect, so they do not have any common point.

Hence, it is a null set.

Question 2 Which of the following sets are finite or infinite:

(i) The set of months of a year

(ii) {1, 2, 3, . . .}

(iii) {1, 2, 3, . . .99, 100}

(iv) The set of positive integers greater than 100.

(v) The set of prime numbers less than 99

Answer:

(i) The number of months in a year is 12 and finite.

Hence, this set is finite.

(ii) {1,2,3,4.......} and so on, this does not have any limit.

Hence, this is an infinite set.

(iii) {1,2,3,4,5......100} has finite numbers.

Hence, this is a finite set.

(iv) Positive integers greater than 100 have no limit.

Hence, it is an infinite set.

(v) Prime numbers less than 99 are finite, known numbers.

Hence, it is a finite set.

Question 3 State whether each of the following sets is finite or infinite:

(i) The set of lines which are parallel to the x-axis

(ii) The set of letters in the English alphabet

(iii) The set of numbers which are multiples of 5

(iv) The set of animals living on the earth

(v) The set of circles passing through the origin (0,0)

Answer:

(i) Lines parallel to the x-axis are infinite.

Hence, it is an infinite set.

(ii) Letters in the English alphabet are 26 finite letters.

Hence, it is a finite set.

(iii) Numbers which are multiples of 5 have no limit; they are infinite.

Hence, it is an infinite set.

(iv) Animals living on earth are finite, though the number is very high.

Hence, it is a finite set.

(v) There is an infinite number of circles which pass through the origin.

Hence, it is an infinite set.

Question 4(i) In the following, state whether A = B or not:

A = {a, b, c, d} B = {d, c, b, a}

Answer:

Given
A = {a, b, c, d}

B = {d, c, b, a}
Comparing the elements of set A and set B, we conclude that all the elements of A and all the elements of B are equal.
Hence, A = B.

Question 4(ii) In the following, state whether A = B or not:

A = {4, 8, 12, 16} B = {8, 4, 16, 18}

Answer:

12 belongs to A, but 12 does not belong to B

12 $\in$ A but 12 $\notin$ B.

Hence, A $\ne$ B.

Question 4(iii) In the following, state whether A = B or not:

A = {2, 4, 6, 8, 10} B = {x: x is positive even integer and $x\le 10$}

Answer:

Positive even integers less than or equal to 10 are, and 2,4,6,8,10.

So, B = {2,4,6,8,10} which is equal to A = {2,4,6,8,10}

Hence, A = B.

Question 4(iv) In the following, state whether A = B or not:

A = {x: x is a multiple of 10}, B = {10, 15, 20, 25, 30, . . .}

Answer:

Multiples of 10 are 10,20,30,40, ........ till infinitythat and....

So, A = {10,20,30,40, .........}

B = {10,15,20,25,30........}

Comparing elements of A and B, we conclude that elements of A and B are not equal.

Hence, A $\ne$ B.

Question 5(i) Are the following pair of sets equal? Give reasons.

A = {2, 3}, B = {x: x is solution of $x^2$ + 5x + 6 = 0}

Answer:

As given,

A = {2,3}

And,

$x^2+5x+6=0$

$x(x+3)+2(x+3)=0$

(x+2)(x+3)=0

x = -2 and -3

B = {-2, -3}

Comparing elements of A and B, we conclude that elements of A and B are not equal.

Hence, A $\ne$ B.

Question 5(ii) Are the following pair of sets equal? Give reasons.

A = {x: x is a letter in the word FOLLOW}

B = {y: y is a letter in the word WOLF}

Answer:

Letters of the word FOLLOW are F, OL, and W.

SO, A = {F, O, L, W}

Letters of the word WOLF are W, O, L, and F.

So, B = {W, O, L, F}

Comparing A and B, we conclude that the elements of A are equal to the elements of B.

Hence, A=B.

Question 6 From the sets given below, select equal sets :

A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2}

E = {–1, 1}, F = {0, a}, G = {1, –1}, H = {0, 1}

Answer:

Compare the elements of A, B, C, D, E, F, G, and H.

8 $\in$ A but $8\notin B,8\in C,8\notin D,8\notin E,8\notin F,8\notin G,8\notin H$

Now, 2 $\in$ A but 2 $\notin$ C.

Hence, A $\ne$ B, A $\ne$ C, A $\ne$ D, A $\ne$ E, A $\ne$ F, A $\ne$ G, A $\ne$ H.

3 $\in$ B,3 $\in$ D but 3 $\notin$ C,3 $\notin$ E,3 $\notin$ F,3 $\notin$ G,3 $\notin$ H.

Hence,B $\ne$ C,B $\ne$ E,B $\ne$ F,B $\ne$ G,B $\ne$ H.

Similarly, comparing other elements of all sets, we conclude that elements of B and D are equal, and elements of E and G are equal.

Hence, B=D and E = G.

NCERT class 11 maths chapter 1 question answer sets - Exercise: 1.3 (Page no:12, Total Question 8)

Question 1 Make correct statements by filling in the symbols $\subset$ or $\not\subset$ in the blank spaces :

(i) { 2, 3, 4 } _____ { 1, 2, 3, 4,5 }

(ii) { a, b, c }______ { b, c, d }

(iii) {x : x is a student of Class XI of your school}______{x : x student of your school}

(iv) {x: x is a circle in the plane} ______{x: x is a circle in the same plane with radius 1 unit}

(v) {x : x is a triangle in a plane} ______ {x : x is a rectangle in the plane}

(vi) {x : x is an equilateral triangle in a plane} ______{x : x is a triangle in the same plane}

(vii) {x : x is an even natural number} _____ {x : x is an integer}

Answer:

A set A is said to be a subset of a set B if every element of A is also an element of B.

(i). All elements {2,3,4} are also elements of {1,2,3,4,5} .

So, {2,3,4} $\subset$ {1,2,3,4,5}.

(ii) All elements { a, b, c } are not elements of{ b, c, d }.

Hence, { a, b, c } $\not\subset$ { b, c, d }.

(iii) Students of class XI are also students of your school.

Hence,{x : x is a student of Class XI of your school} $\subset$ {x : x student of your school}

(iv). Here, {x: x is a circle in the plane} $\not\subset$ {x: x is a circle in the same plane with radius 1 unit}: since a circle in the plane can have any radius

(v). Triangles and rectangles are two different shapes.

Hence,{x : x is a triangle in a plane} $\not\subset$ {x : x is a rectangle in the plane}

(vi). Equilateral triangles are part of all types of triangles.

So,{x : x is an equilateral triangle in a plane} $\subset$ {x : x is a triangle in the same plane}

(vii). Even natural numbers are part of all integers.

Hence, {x : x is an even natural number} $\subset$ {x : x is an integer}

Question 2 Examine whether the following statements are true or false:

(i) { a, b } $\not\subset$ { b, c, a }

(ii) { a, e } $\subset$ { x : x is a vowel in the English alphabet}

(iii) { 1, 2, 3 } $\subset$ { 1, 3, 5 }

(iv) { a } $\subset$ { a, b, c }

(v) { a } $\in$ { a, b, c }

(vi) { x : x is an even natural number less than 6} $\subset$ { x : x is a natural number which divides 36}

Answer:

(i) All elements of { a, b } lie in { b, c, a }.So,{ a, b } $\subset$ { b, c, a }.

Hence, it is false.

(ii) All elements of { a, e } lie in {a,e,i,o,u}.

Hence, the statement is true.

(iii) All elements of { 1, 2, 3 } are not present in { 1, 3, 5 }.

Hence, the statement given is false.

(iv) Element of { a } lie in { a, b, c }.

Hence, the statement is true.

(v). { a } $\subset$ { a, b, c }

So, the statement is false.

(vi) All elements {2,4,} lies in {1,2,3,4,6,9,12,18,36}.

Hence, the statement is true.

Question 3(i) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{3, 4} $\subset$ A

Answer:

3 $\in$ {3,4} but 3 $\notin$ {1,2,{3,4},5}.

SO, {3, 4} $\not\subset$ A

Hence, the statement is incorrect.

Question 3(ii) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{3, 4} $\in$ A

Answer:

{3, 4} is an element of A.

So, {3, 4} $\in$ A.

Hence, the statement is correct.

Question 3(iii) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{{3, 4}} $\subset$ A

Answer:

Here,

{ 3, 4 } $\in$ { 1, 2, { 3, 4 }, 5 }

and { 3, 4 } $\in$ {{3, 4}}

So, {{3, 4}} $\subset$ A.

Hence, the statement is correct.

Question 3(iv) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

1 $\in$ A

Answer:

Given, 1 is element of { 1, 2, { 3, 4 }, 5 }.

So,1 $\in$ A.

Hence, the statement is correct.

Question 3(v) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

1 $\subset$ A

Answer:

Here, 1 is an element of set A = { 1, 2, { 3, 4 }, 5 }. So, elements of set A cannot be a subset of set A.

1 $\not\subset$ { 1, 2, { 3, 4 }, 5 }.

Hence, the statement given is incorrect.

Question 3(vi) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{1,2,5} $\subset$ A

Answer:

All elements of {1,2,5} are present in { 1, 2, { 3, 4 }, 5 }.

So, {1,2,5} $\subset$ { 1, 2, { 3, 4 }, 5 }.

Hence, the statement given is correct.

Question 3(vii) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{1,2,5} $\in$ A

Answer:

Here,{1,2,5} is not an element of { 1, 2, { 3, 4 }, 5 }.

So,{1,2,5} $\notin$ A .

Hence, the statement is incorrect.

Question 3(viii) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{1,2,3} $\subset$ A

Answer:

Here, 3 $\in$ {1,2,3}

but 3 $\notin$ { 1, 2, { 3, 4 }, 5 }.

So, {1,2,3} $\not\subset$ A

Hence, the given statement is incorrect.

Question 3(ix) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

$\varphi \in A$

Answer:

$\varphi$ is not an element of { 1, 2, { 3, 4 }, 5 }.

So, $\varphi \notin A$ .

Hence, the above statement is incorrect.

Question 3(x) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

$\varphi \subset A$

Answer:

$\varphi$ is a subset of all sets.

Hence, the above statement is correct.

Question 3(xi) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

$\left\{\varphi \right\}\subset A$

Answer:

$\varphi$ $\subset$ { 1, 2, { 3, 4 }, 5 }. but $\varphi$ is not an element of { 1, 2, { 3, 4 }, 5 }.

$\left\{\varphi \right\}\not\subset A$

Hence, the above statement is incorrect.

Question 4(i) Write down all the subsets of the following sets

{a}

Answer:

Subsets of $\left\{a\right\}=\varphi and\left\{a\right\}$ .

Question 4 (ii) Write down all the subsets of the following sets:

{a, b}

Answer:

Subsets of $\{a,b\}are\varphi ,\left\{a\right\},\left\{b\right\}and\{a,b\}$ . Thus, the given set has 4 subsets

Question 4 (iii) Write down all the subsets of the following sets:

{1,2,3}

Answer:

Subsets of

$\{1,2,3\}=\left\{1\right\},\left\{2\right\},\left\{3\right\},\varphi ,\{1,2\},\{2,3\},\{3,1\},\{1,2,3\}$

Question 4 (iv) Write down all the subsets of the following sets:

$\varphi$

Answer:

Subset of $\varphi$ is $\varphi$ only.

The subset of a null set is the null set itself.

Question 5 Write the following as intervals :

(i) {x : x $\in$ R, – 4 $<$ x $\le$ 6}

(ii) {x : x $\in$ R, – 12 $<$ x $<$ –10}

(iii) {x : x $\in$ R, 0 $\le$ x $<$ 7}

(iv) {x : x $\in$ R, 3 $\le$ x $\le$ 4}

Answer:

The following can be written in the interval as :

(i) {x : x $\in$ R, – 4 $<$ x $\le$ 6} $=(-4,6]$

(ii) {x : x $\in$ R, – 12 $<$ x $<$ –10} $=(-12,-10)$

(iii) {x : x $\in$ R, 0 $\le$ x $<$ 7}=[0,7)

(iv) (iv) {x : x $\in$ R, 3 $\le$ x $\le$ 4} $=[3,4]$

Question 6 Write the following intervals in set-builder form :

(i) (– 3, 0)

(ii) [6, 12]

(iii) (6, 12]

(iv) [–23, 5)

Answer:

The given intervals can be written in set builder form as :

(i) (– 3, 0) $=\{x:x\in R,-3<x<0\}$

(ii) [6 , 12] $=\{x:x\in R,6\le x\le 12\}$

(iii) (6, 12] $=\{x:x\in R,6<x\le 12\}$

(iv) [–23, 5) $=\{x:x\in R,-23\le x<5\}$

Question 7 What universal set(s) would you propose for each of the following :

(i) The set of right triangles.

(ii) The set of isosceles triangles.

Answer:

(i) The universal set for a set of right-angle triangles can be a set of polygons or a set of all triangles.

(ii) The universal set for a set of isosceles angle triangles can be a set of polygons.

Question 8 should be added.

NCERT class 11 maths chapter 1 question answer sets - Exercise: 1.4 (Page no: 17, Total Question: 12)

Question 1(i) Find the union of each of the following pairs of sets :

X = {1, 3, 5} Y = {1, 2, 3}

Answer:

Union of X and Y is X $\cup$ Y = {1,2,3,5}

Question 1 (ii) Find the union of each of the following pairs of sets :

A = [ a, e, i, o, u} B = {a, b, c}

Answer:

Union of A and B is A $\cup$ B = {a, b, c, e, i, o, u}.

Question 1 (iii) Find the union of each of the following pairs of sets :

A = {x : x is a natural number and multiple of 3}, B = {x : x is a natural number less than 6}

Answer:

Here,

A = {3,6,9,12,15,18, ............}

B = {1,2,3,4,5,6}

The Union of A and B is A $\cup$ B.

A $\cup$ B = {1,2,3,4,5,6,9,12,15........}

Question 1 (iv) Find the union of each of the following pairs of sets :

A = {x: x is a natural number and 1 $<$ x $\le$ 6}

B = {x: x is a natural number and 6 $<$ x $<$ 10}

Answer:

Here,

A = {2,3,4,5,6}

B = {7,8,9}

A $\cup$ B = {2,3,4,5,6,7,8,9}

or it can be written as A $\cup$ B = {x : x is a natural number and 1 $<$ x $<$ 10 }

Question 1 (v) Find the union of each of the following pairs of sets :

A = {1, 2, 3} B = $\varphi$

Answer:

Here,

A union B is A $\cup$ B.

A $\cup$ B = {1,2,3}

Question 2 Let A = { a, b }, B = {a, b, c}. Is A $\subset$ B ? What is A $\cup$ B?

Answer:

Here,

We can see elements of A lie in set B.

Hence, A $\subset$ B.

And, A $\cup$ B = {a,b,c} = B

Question 3 If A and B are two sets such that A $\subset$ B, then what is A $\cup$ B?

Answer:

If A is a subset of B, then A $\cup$ B will be set B.

Question 4(i) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

$A\cup B$

Answer:

Here,

A = {1, 2, 3, 4}

B = {3, 4, 5, 6}

The union of the set can be written as follows

A $\cup$ B = {1,2,3,4,5,6}

Question 4(ii) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

$A\cup C$

Answer:

Here,

A = {1, 2, 3, 4}

C = {5, 6, 7, 8}

The union can be written as follows

A $\cup$ C = {1,2,3,4,5,6,7,8}

Question 4(iii) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

B $\cup$ C

Answer:

Here,

B = {3, 4, 5, 6},

C = {5, 6, 7, 8}

The union of the given sets are

B $\cup$ C = {3,4,5,6,7,8}

Question 4(iv) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

B $\cup$ D

Answer:

Here,

B = {3, 4, 5, 6}

D = {7, 8, 9, 10}

B $\cup$ D = {3,4,5,6,7,8,9,10}

Question 4(v) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

A $\cup$ B $\cup$ C

Answer:

Here,

A = {1, 2, 3, 4},

B = {3, 4, 5, 6},

C = {5, 6, 7, 8}

The union can be written as

A $\cup$ B $\cup$ C = {1,2,3,4,5,6,7,8}

Question 4(vi) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

A $\cup$ B $\cup$ D

Answer:

Here,

A = {1, 2, 3, 4},

B = {3, 4, 5, 6}

D = {7, 8, 9, 10}

The union can be written as

A $\cup$ B $\cup$ D = {1,2,3,4,5,6,7,8,9,10}

Question 4(vii) If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

B $\cup$ C $\cup$ D

Answer:

Here, B = {3, 4, 5, 6},

C = {5, 6, 7, 8 } and

D = { 7, 8, 9, 10 }

The union can be written as

B $\cup$ C $\cup$ D = {3,4,5,6,7,8,9,10}

Question Find the intersection of each pair of sets of question 1 above

(i) X = {1, 3, 5} Y = {1, 2, 3}

(ii) A = [ a, e, i, o, u} B = {a, b, c}

(iii) A = {x : x is a natural number and multiple of 3} B = {x : x is a natural number less than 6}

(iv) A = {x : x is a natural number and 1 $<$ x $\le$ 6 } B = {x : x is a natural number and 6 $<$ x $<$ 10 }

(v) A = {1, 2, 3}, B = $\varphi$

Answer:

(i) X = {1, 3, 5} Y = {1, 2, 3}

X $\cap$ Y = {1,3}

(ii) A = [ a, e, i, o, u} B = {a, b, c}

A $\cap$ B = {a}

(iii) A = {x : x is a natural number and multiple of 3} B = {x : x is a natural number less than 6}

A = {3,6,9,12,15.......} B = {1,2,3,4,5}

A $\cap$ B = {3}

(iv)A = {x : x is a natural number and 1 $<$ x $\le$ 6 } B = {x : x is a natural number and 6 $<$ x $<$ 10 }

A = {2,3,4,5,6} B = {7,8,9}

A $\cap$ B = $\varphi$

(v) A = {1, 2, 3}, B = $\varphi$

A $\cap$ B = $\varphi$

Question 6 If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; find

(i) A $\cap$ B (ii) B $\cap$ C

(iii) A $\cap$ C $\cap$ D (iv) A $\cap$ C

(v) B $\cap$ D (vi) A $\cap$ (B $\cup$ C)

(vii) A $\cap$ D (viii) A $\cap$ (B $\cup$ D)

(ix) ( A $\cap$ B ) $\cap$ ( B $\cup$ C )

(x) ( A $\cup$ D) $\cap$ ( B $\cup$ C)

Answer:

Here, A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}

(i) A $\cap$ B = {7,9,11} (vi) A $\cap$ (B $\cup$ C) = {7,9,11}

(ii) B $\cap$ C = { 11,13} (vii) A $\cap$ D = $\varphi$

(iii) A $\cap$ C $\cap$ D = $\varphi$ (viii) A $\cap$ (B $\cup$ D) = {7,9,11}

(iv) A $\cap$ C = { 11 } (ix) ( A $\cap$ B ) $\cap$ ( B $\cup$ C ) = {7,9,11}

(v) B $\cap$ D = $\varphi$ (x) ( A $\cup$ D) $\cap$ ( B $\cup$ C) = {7,9,11,15}

Question 7 If A = {x : x is a natural number }, B = {x : x is an even natural number} C = {x : x is an odd natural number} and D = {x : x is a prime number }, find

(i) A $\cap$ B

(ii) A $\cap$ C

(iii) A $\cap$ D

(iv) B $\cap$ C

(v) B $\cap$ D

(vi) C $\cap$ D

Answer:

Here, A = {1,2,3,4,5,6...........}

B = {2,4,6,8,10...........}

C = {1,3,5,7,9,11,...........}

D = {2,3,5,7,11,13,17,......}

(i) A $\cap$ B = {2,4,6,8,10........} = B

(ii) A $\cap$ C = {1,3,5,7,9.........} = C

(iii) A $\cap$ D = {2,3,5,7,11,13.............} = D

(iv) B $\cap$ C = $\varphi$

(v) B $\cap$ D = {2}

(vi) C $\cap$ D = {3,5,7,11,13,..........} = $(x:xisoddprimenumber)$

Question 8(i) Which of the following pairs of sets are disjoint

{1, 2, 3, 4} and {x : x is a natural number and 4 $\le$ x $\le$ 6 }

Answer:

Here, {1, 2, 3, 4} and {4,5,6}

{1, 2, 3, 4} $\cap$ {4,5,6} = {4}

Hence, it is not a disjoint set.

Question 8(ii) Which of the following pairs of sets are disjoint

{ a, e, i, o, u } and { c, d, e, f }

Answer:

Here, { a, e, i, o, u } and { c, d, e, f }

{ a, e, i, o, u } $\cap$ { c, d, e, f } = {e}

Hence, it is not a disjoint set.

Question 8(iii) Which of the following pairs of sets are disjoint

{x : x is an even integer } and {x : x is an odd integer}

Answer:

Here, {x : x is an even integer } and {x : x is an odd integer}

{2,4,6,8,10,..........} and {1,3,5,7,9,11,.....}

{2,4,6,8,10,..........} $\cap$ {1,3,5,7,9,11,.....} = $\varphi$

Hence, it is a disjoint set.

Question 9 If A = {3, 6, 9, 12, 15, 18, 21}, B = { 4, 8, 12, 16, 20 }, C = { 2, 4, 6, 8, 10, 12, 14, 16 }, D = {5, 10, 15, 20 }; find

(i) A – B (ii) A – C (iii) A – D (iv) B – A (v) C – A (vi) D – A

(vii) B – C (viii) B – D (ix) C – B (x) D – B (xi) C – D (xii) D – C

Answer:

A = {3, 6, 9, 12, 15, 18, 21}, B = { 4, 8, 12, 16, 20 }, C = { 2, 4, 6, 8, 10, 12, 14, 16 }, D = {5, 10, 15, 20 }

The given operations are done as follows

(i) A – B = {3,6,9,15,18,21} (vii) B – C = {20}

(ii) A – C = {3,9,15,18,21} (viii) B – D = {4,8,12,16}

(iii) A – D = {3,6,9,12,18,21} (ix) C – B = {2,6,10,14}

(iv) B – A = {4,8,16,20} (x) D – B = {5,10,15}

(v) C – A = {2,4,8,10,14,16} (xi) C – D = {2,4,6,8,12,14,16}

(vi) D – A = {5,10,20} (xii) D – C = {5,15,20}

Question 10 If X= { a, b, c, d } and Y = { f, b, d, g}, find

(i) X – Y

(ii) Y – X

(iii) X $\cap$ Y

Answer:

X= { a, b, c, d } and Y = { f, b, d, g}

(i) X – Y = {a,c}

(ii) Y – X = {f,g}

(iii) X $\cap$ Y = {b,d}

Question 11 : If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q ?

Answer:

R = set of real numbers.

Q = set of rational numbers.

R - Q = set of irrational numbers.

Question 12 (i) State whether each of the following statements is true or false. Justify your answer.

{ 2, 3, 4, 5 } and { 3, 6} are disjoint sets.

Answer:

Here,

{ 2, 3, 4, 5 } and { 3, 6}

{ 2, 3, 4, 5 } $\cap$ { 3, 6} = {3}

Hence, these are not disjoint sets.

So, false.

Question 12 (ii) State whether each of the following statements about the three dimensions is true or false. Justify your answer.

{ a, e, i, o, u } and { a, b, c, d }are disjoint sets

Answer:

Here, { a, e, i, o, u } and { a, b, c, d }

{ a, e, i, o, u } $\cap$ { a, b, c, d } = {a}

Hence, these are not disjoint sets.

So, the statement is false.

Question 12 (iii) State whether each of the following statements is true or false. Justify your answer.

{ 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.

Answer:

Here,

{ 2, 6, 10, 14 } and { 3, 7, 11, 15}

{ 2, 6, 10, 14 } $\cap$ { 3, 7, 11, 15} = $\varphi$

Hence, these are disjoint sets.

So, given the statement is true.

Question 12 (iv) State whether each of the following statements is true or false. Justify your answer.

{ 2, 6, 10 } and { 3, 7, 11} are disjoint sets.

Answer:

Here,

{ 2, 6, 10 } and { 3, 7, 11}

{ 2, 6, 10 } $\cap$ { 3, 7, 11} = $\varphi$

Hence, these are disjoint sets.

So, the statement is true.

Class 11 maths chapter 1 NCERT solutions sets-Exercise: 1.5 (Page no: 20, Total Question )

Question 1 Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 }. Find

(i) A′

(ii) B′

(iii) (A $\cup$ C)′

(iv) (A $\cup$ B)′

(v) (A')'

(vi) (B – C)'

Answer:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 }

(i) A′ = U - A = {5,6,7,8,9}

(ii) B′ = U - B = {1,3,5,7,9}

(iii) A $\cup$ C = {1,2,3,4,5,6}

(A $\cup$ C)′ = U - (A $\cup$ C) = {7,8,9}

(iv) (A $\cup$ B) = {1,2,3,4,6,8}

(A $\cup$ B)′ = U - (A $\cup$ B) = {5,7,9}

(v) (A')' = A = { 1, 2, 3, 4}

(vi) (B – C) = {2,8}

(B – C)' = U - (B – C) = {1,3,4,5,6,7,9}

Question 2 If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :

(i) A = {a, b, c}

(ii) B = {d, e, f, g}

(iii) C = {a, c, e, g}

(iv) D = { f, g, h, a}

Answer:

U = { a, b, c, d, e, f, g, h}

(i) A = {a, b, c}

A' = U - A = {d,e,f,g,h}

(ii) B = {d, e, f, g}

B' = U - B = {a,b,c,h}

(iii) C = {a, c, e, g}

C' = U - C = {b,d,f,h}

(iv) D = { f, g, h, a}

D' = U - D = {b,c,d,e}

Question 3: Taking the set of natural numbers as the universal set, write down the complements of the following sets:

(i) {x : x is an even natural number}

(ii) { x : x is an odd natural number }

(iii) {x : x is a positive multiple of 3}

(iv) { x : x is a prime number }

(v) {x: x is a natural number divisible by 3 and 5}

Answer:

Universal set = U = {1,2,3,4,5,6,7....................}

(i) {x : x is an even natural number} = {2,4,6,8,..........}

{x : x is an even natural number}'= U - {x : x is an even natural number} = {1,3,5,7,9,..........} = {x : x is an odd natural number}

(ii) { x : x is an odd natural number }' = U - { x : x is an odd natural number } = {x : x is an even natural number}

(iii) {x : x is a positive multiple of 3}' = U - {x : x is a positive multiple of 3} = {x : x , x $\in$ N and is not a positive multiple of 3}

(iv) { x : x is a prime number }' = U - { x : x is a prime number } = { x : x is a positive composite number and 1 }

(v) {x : x is a natural number divisible by 3 and 5}' = U - {x : x is a natural number divisible by 3 and 5} = {x : x is a natural number not divisible by 3 or 5}

Question 3: Taking the set of natural numbers as the universal set, write down the complements of the following sets:

(vi) { x : x is a perfect square }

(vii) { x : x is a perfect cube}

(viii) { x : x + 5 $=$ 8 }

(ix) { x : 2x + 5 $=$ 9}

(x) { x : x $\ge$ 7 }

(xi) { x : x $\in$ N and 2x + 1 $>$ 10 }

Answer:

Universal set = U = {1,2,3,4,5,6,7,8.............}

(vi) { x : x is a perfect square }' = U - { x : x is a perfect square } = { x : x $\in$ N and x is not a perfect square }

(vii) { x : x is a perfect cube}' = U - { x : x is a perfect cube} = { x : x $\in$ N and x is not a perfect cube}

(viii) { x : x + 5 $=$ 8 }' = U - { x : x + 5 $=$ 8 } = U - {3} = { x : x $\in$ N and x $\ne$ 3 }

(ix) { x : 2x + 5 $=$ 9}' = U - { x : 2x + 5 $=$ 9} = U -{2} = { x : x $\in$ N and x $\ne$ 2}

(x) { x : x $\ge$ 7 }' = U - { x : x $\ge$ 7 } = { x : x $\in$ N and x $<$ 7}

(xi) { x : x $\in$ N and 2x + 1 $>$ 10 }' = U - { x : x $\in$ N and x $>$ 9/2 } = { x : x $\in$ N and x $\le$ 9/2 }

Question 4 If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that

(i) (A $\cup$ B)′ = A′ $\cap$ B′

(ii) (A $\cap$ B)′ = A′ $\cup$ B′

Answer:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}

(i) (A $\cup$ B)′ = A′ $\cap$ B′

L.H.S = (A $\cup$ B)′ = U - (A $\cup$ B) = {1,9}

R.H.S = A′ $\cap$ B′ = {1,3,5,7,9} $\cap$ {1,4,6,8,9} = {1,9}

L.H.S = R.H.S

Hence, the statement is true.

(ii) (A $\cap$ B)′ = A′ $\cup$ B′

L.H.S = U - (A $\cap$ B) ={1,3,4,5,6,7,8,9}

R.H.S = A′ $\cup$ B′ = {1,3,5,7,9} $\cup$ {1,4,6,8,9} = {1,3,4,5,6,7,8,9}

L.H.S = R.H.S

Hence, the statement is true.

Question 5 : Draw an appropriate Venn diagram for each of the following :

(i) (A ∪ B)′

(ii) A′ \cap B′

(iii) (A \cap B)′

(iv) A′ ∪ B′

Answer:

(i) $(A\cup B{)}^{\prime }$

$(A\cup B)$ is in the yellow colour

$(A\cup B{)}^{\prime }$ is in green colour

ii) $A\prime \cap B\prime$ is represented by the green colour in the below figure

iii) $(A\cap B)\prime$ is represented by green colour in the below diagram and white colour represents $(A\cap B)$

iv) $A^{\prime }\cup B^{\prime }$

The green colour represents $A^{\prime }\cup B^{\prime }$.

Question 6 Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from ${60}^{\circ }$, what is A′?

Answer:

$A^{\prime }$ is the set of all triangles whose angle is ${60}^{\circ }$, in other words, $A^{\prime }$ is the set of all equilateral triangles.

Question 7 Fill in the blanks to make each of the following a true statement :

(i) A $\cup$ A′ $=$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

(ii) ${\varphi }^{\prime }$ $\cap$ A $=$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

(iii) A $\cap$ A′ $=$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

(iv) U′ $\cap$ A $=$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot$

Answer:

The following are the answers to the questions

(i) A $\cup$ A′ $=$ U

(ii) $\varphi$ ′ $\cap$ A $=$ A

(iii) A $\cap$ A′ $=$ $\varphi$

(iv) U′ $\cap$ A $=$ $\varphi$

NCERT class 11 maths chapter 1 question answer sets - Miscellaneous Exercise (Page no: 21, Total Question)

Question 1 Decide, among the following sets, which sets are subsets of one another:

A = { x : x $\in$ R and x satisfy $x^2$ – 8x + 12 = 0 }, B = { 2, 4, 6 },

C = { 2, 4, 6, 8, . . . }, D = { 6 }.

Answer:

Solution of this equation are $x^2$ – 8x + 12 = 0

( x - 2 ) ( x - 6 ) = 0

X = 2,6

$\therefore$ A = { 2,6 }

B = { 2, 4, 6 }

C = { 2, 4, 6, 8, . . . }

D = { 6 }

From the sets given above, we can conclude that A $\subset$ B, A $\subset$ C, D $\subset$ A, D $\subset$ B, D $\subset$ C, B $\subset$ C.

Hence, we can say that D $\subset$ A $\subset$ B $\subset$ C

Question 2(i) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If x $\in$ A and A $\in$ B , then x $\in$ B

Answer:

The given statement is false,

example: Let A = { 2,4 }

B = { 1,{2,4},5}

x be 2.

then, 2 $\in$ { 2,4 } = x $\in$ A and { 2,4 } $\in$ { 1,{2,4},5} = A $\in$ B

But 2 $\notin$ { 1,{2,4},5} i.e. x $\notin$ B

Question 2(ii) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A $\subset$ B and B $\in$ C , then A $\in$ C

Answer:

The given statement is false,

Let, A = {1}

B = {1,2,3}

C = {0, {1,2,3},4}

Here, {1} $\subset$ {1,2,3} = A $\subset$ B and {1,2,3} $\in$ {0,{1,2,3},4} = B $\in$ C

But, {1} $\notin$ {0,{1,2,3},4} = A $\notin$ C

Question 2(iii) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A $\subset$ B and B $\subset$ C , then A $\subset$ C

Answer:

Let A ⊂ B and B ⊂ C

There is an element x such that

Let, x $\in$ A

$\Rightarrow$ x $\in$ B (Because A $\subset$ B)

$\Rightarrow$ x $\in$ C (Because B $\subset$ C)

Hence, the statement is true that A $\subset$ C.

Question 2(iv) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A $\not\subset$ B and B $\not\subset$ C, then A $\not\subset$ C

Answer:

The given statement is false

Let, A = {1,2}

B = {3,4,5}

C = {1,2,6,7,8}

Here, {1,2} $\not\subset$ {3,4,5} = A $\not\subset$ B and {3,4,5} $\not\subset$ {1,2,6,7,8} = B $\not\subset$ C

But, {1,2} $\subset$ {1,2,6,7,8} = A $\subset$ C

Question 2 (v) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If x $\in$ A and A $\not\subset$ B , then x $\in$ B

Answer:

The given statement is false,

Let x be 2

A = {1,2,3}

B = {4,5,6,7}

Here, 2 $\in$ {1,2,3} = x $\in$ A and {1,2,3} $\not\subset$ {4,5,6,7} = A $\not\subset$ B

But, 2 $\notin$ {4,5,6,7} implies x $\notin$ B

Question 2 (vi) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A $\subset$ B and x $\notin$ B, then x $\notin$ A

Answer:

The given statement is true,

Let, A $\subset$ B and x $\notin$ B

Suppose, x $\in$ A

Then, x $\in$ B , which is contradiction to x $\notin$ B

Hence, x $\notin$ A.

Question 3 Let A, B, and C be the sets such that A $\cup$ B = A $\cup$ C and A $\cap$ B = A $\cap$ C. Show that B = C.

Answer:

Let A, B, and C be the sets such that A $\cup$ B = A $\cup$ C and A $\cap$ B = A $\cap$ C

To prove: B = C.

A $\cup$ B = A + B - A $\cap$ B = A $\cup$ C = A + C - A $\cap$ C

A + B - A $\cap$ B = A + C - A $\cap$ C

B - A $\cap$ B = C - A $\cap$ C ( since A $\cap$ B = A $\cap$ C )

B = C

Hence, it is proved that B = C.

Question 4 Show that the following four conditions are equivalent :

(i) A $\subset$ B (ii) A – B = $\varphi$ (iii) A $\cup$ B = B (iv) A $\cap$ B = A

Answer:

First, we need to show A $\subset$ B $\iff$ A – B = $\varphi$

Let A $\subset$ B

To prove: A – B = $\varphi$

Suppose A – B $\ne \varphi$

this means, x $\in$ A and x $\ne$ B , which is not possible as A $\subset$ B .

So, A – B = $\varphi$.

Hence, A $\subset$ B $⟹$ A – B = $\varphi$ .

Now, let A – B = $\varphi$

(i). To prove: A $\subset$ B

Suppose, x $\in$ A

A – B = $\varphi$ so x $\in$ B

Since, x $\in$ A and x $\in$ B and A – B = $\varphi$ so A $\subset$ B

Hence, A $\subset$ B $\iff$ A – B = $\varphi$.

Let A $\subset$ B

(ii). To prove: A $\cup$ B = B

We can say B $\subset$ A $\cup$ B

Suppose, x $\in$ A $\cup$ B

means x $\in$ A or x $\in$ B

If x $\in$ A

since A $\subset$ B so x $\in$ B

Hence, A $\cup$ B = B

and If x $\in$ B then also A $\cup$ B = B.

Now, let A $\cup$ B = B

(iii). To prove: A $\subset$ B

Suppose: x $\in$ A

A $\subset$ A $\cup$ B so x $\in$ A $\cup$ B

A $\cup$ B = B so x $\in$ B

Hence, A $\subset$ B

Also, A $\subset$ B $\iff$ A $\cup$ B = B