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1.Sets, Relation and Functions
- Introduction to sets, Description of sets
- Types of Sets, Subsets
- Intervals, Venn Diagrams, Operations on Sets
- Laws of Algebra of Sets
- Introduction to sets and its types, operations of sets, Venn Diagrams
- Functions and its Types
- Functions Types
- Cartesian Product of Sets, Relation, Domain and Range
- Sum Related to Relations
- Sums Related to Relations, Domain and Range
- Chapter Notes – Sets, Relation and Functions
- NCERT Solutions – Sets, Relation and Functions
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2.Trigonometric Functions
- Introduction, Some Identities and Some Sums
- Some Sums Related to Trigonometry Identities, trigonometry Functions Table and Its Quadrants
- NCERT Sums Ex.3.3 (Q.1-5)Based on Trigometry table and Their Quadrants, Trigonometry Identities of Sum and Diff. of two Angles
- NCERT Sums Ex-3.2 Based on Trigonometry Function of Lower & Higher Angles
- NCERT Sums Ex-3.3 (Q.6 – 10) Based on Radian Angles
- NCERT Sums Ex-3.3 (Q.11-13)Based on Trigonometry Identities
- NCERT Sums Ex-3.3 (Q. 14)Based on Trigonometry Identities
- NCERT Sums Ex-3.3 (Q.16) Based on Trigonometry Identities
- NCERT Sums Ex-3.3 (Q.17 -21) Based on Trigonometry Identities
- NCERT Sums Ex-3.4 (Q. 1 – 9), Trigonometry Equation
- Sums Based on Trigonometry Equations
- Sums Based on Trigonometry Equations
- Sums Based on Trigonometry Equations
- Sums Based on Trigonometry Equations
- Equations Having two Variable Angle which satisfy both equations
- Trigonometrical Identities-Some important relations and Its related Sums
- Sums Related to Trigonometrical Identities
- Properties of Triangles and Solution of Triangles-Sine formula, Napier Analogy and Sums
- Relation Between Degree and Radian, Quadrant and NCERT Sum Ex.3.1, 3.2
- Trigonometric Functions Table
- Some Trigonometric Identities and its related Sums
- Sums Related to Trigonometrical Identities
- Sums Related to Trigonometrical Identities
- Sums Related to Trigonometrical Identities
- Trigonometry Equations
- Sum Based on Trigonometry Equations
- Sums Based on Trigonometry function of Lower Angle
- Chapter Notes – Trigonometric Functions
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3.Mathematical Induction
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4.Complex Numbers and Quadratic Equation
- Introduction, Nature of Roots, Numbers, Introduction of i
- Sum Related to Relations, Real and Imaginary part of C-N, Conjugate of a C-N
- Absolute value or Modulus of a C-N and Related Sums
- Sums Related To Multiplicative Inverse
- Polar Form of a C-N
- Sums Related To Polar Form
- Square Roots of C-N and its Related Sums
- De Moivris Theorem and its related Sums
- Introduction, Nature of Roots, Numbers, Introduction of i and its Sums, Real and Imaginary Part of C-N
- Sums Related to Real and Imaginary Part of C-N and Operations on C-N
- Sums Related To Multiplicative Inverse
- Sums Related To Multiplicative Inverse and Modulus and Argument of a C-N
- Polar form of a C-N, Nature of Roots
- Sums Based on Roots of Quadratic Equations, Sums of Polar form
- Chapter Notes – Complex Numbers and Quadratic Equation
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5.Linear Inequalities
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6.Permutations and Combinations
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7.Binomial Theorem
- Introduction to Binomial Theorem
- Binomial General Expansion and Their Derivations and its Related Sums
- Pascal’s Triangle Theorem, Addition of Two Expansion, NCERT Sums Ex-8.1
- Sums of Miscellaneous Exercise and Ex-8.1, Finding the Any Term from nth Term
- NCERT Sums Ex-8.1
- NCERT Sums Ex-8.1
- NCERT Sums Ex-8.2, Middle Term
- NCERT Sums Ex-8.2, Middle Term Related Sums
- To Find the Coefficient of X^r in the Expansion of (X+A)^n, NCERT Sums Ex-8.2 and Miscellaneous Ex.
- NCERT Sums Ex-8.2
- To Find the Sum of the Coefficients in the Expansion of (1+x)^n and its Related Sums
- Sums Related to Binomials Coefficients
- Binomial Theorem for any Index and its Related Sums
- Introduction to Binomial Theorem, General Term in the Expansion of (x+a)^n.
- NCERT Sums Ex-8.1 & 8.2, Pascals’ Triangle, pth Term from End
- Sums related to Finding the Coefficient, NCERT Sums Ex-8.2, Middle Term
- Sums Related to Middle Term
- Sums Related to Coefficient of the Any Term
- Chapter Notes – Binomial Theorem
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8.Sequences and Series
- Introduction, A.P., nth Term and Sum of nth Term, P Arithmetic Mean B/w a and b, Sum Based on Fibonacci Sequence
- NCERT Sums Ex-9.2
- NCERT Sums Ex-9.2
- NCERT Sums Ex-9.2, Geometric Progression -Introduction, nth term, NCERT Sums Ex-9.3
- NCERT Sums Ex-9.3
- Sum of n term of G.P., NCERT Sums Ex-9.3
- NCERT Sums Ex-9.3
- NCERT Sums Ex-9.3, Insert P Geometrical Mean B/w a and b
- NCERT Sum Ex-9.3
- NCERT Sum Ex-9.3
- Some Special Series, NCERT Sum Ex-9.4
- NCERT Sum Ex-9.4
- NCERT Sum Ex-9.4
- Chapter Notes – Sequences and Series
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9.Properties of Triangles
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10.Straight Lines
- Introduction, Equation of Line, Slope or Gradient of a line
- Sums Related to Finding the Slope, Angle Between two Lines
- Cases for Angle B/w two Lines, Different forms of Line Equation
- Sums Related Finding the Equation of Line
- Sums based on Previous Concepts of Straight line
- Parametric Form of a Straight Line
- Sums Related to Parametric Form of a Straight Line
- Sums Based on Concurrent of lines, Angle b/w Two Lines
- Different condition for Angle b/w two lines
- Sums Based on Angle b/w Two Lines
- Equation of Straight line Passes Through a Point and Make an Angle with Another Line
- Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line
- Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line
- Finding the Distance of a point from the line
- Sum Based on Finding the Distance of a point from the line and B/w Two Parallel Lines
- Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines
- Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines
- Introduction, Distance B/w Two Points, Slope, Equation of Line
- NCERT Sums Ex-10.1
- NCERT Sums Ex-10.1
- NCERT Sums Ex-10.1 & 10.2
- NCERT Sums Ex-10.2
- NCERT Sums Ex-10.2
- NCERT Sums Ex-10.2 & 10.3
- NCERT Sums Ex- 10.3 (Reduce the Equation into intercept Form, Normal form)
- NCERT Sums Ex-10.3
- NCERT Sums Ex-10.3 (Equation of Parallel line, Perpendicular Line of given line, Sums Based of Angle B/w Two Lines)
- NCERT Sums Ex-10.3
- NCERT Sums Ex-10.3
- Chapter Notes – Straight Lines
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11.Conic Sections
- Introduction, General Equation of second Degree, Parabola, Sums based on Finding Equation of Parabola
- Sums Based on Equation of Parabola, Four Forms of Parabola-Form (i)
- Sums Based on Four Forms of Parabola-Form (i)
- Four Forms of Parabola-Form (ii), (iii) (iv)
- Sums Based on Four forms of Parabola
- Position of a Point with Respect to Parabola and its Sums
- Circles-Introduction, Different Cases for Circle Equations, NCERT Sums Ex-11.1
- NCERT Sums Ex-11.1
- Circle Important Point Revise, Intersection of Axes, NCERT Sums Ex-11.1
- NCERT Sums Ex-11.1
- Parabola- Introduction, General Equation , Sums, Some Important Concepts for Parabola
- Different Form of Parabola, NCERT Sum Ex-11.2
- NCERT Sum Ex-11.2
- Ellipse-Introduction, General Equation, NCERT Sums Ex-11.3
- NCERT Sums Ex-11.3
- NCERT Sums Ex-11.3
- Hyperbola-Introduction, NCERT Sums Ex-11.4
- NCERT Sums Ex-11.4
- Chapter Notes – Conic Sections Circles
- Chapter Notes – Conic Sections Ellipse
- Chapter Notes – Conic Sections Parabola
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12.Coordinate Geometry
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13.Three Dimensional Geometry
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14.Limits And Derivatives
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15.Mathematical Reasoning
- What is statement ? Special word and phrases, negation of statement , Compound statement , and & or in compound statement , truth table Solving the problems of Ex- 14.1 , 14.2
- Solving Ex-14.3, Ex-14.4, Implications, Validating statements, Ex-14.5, Direct method
- Chapter Notes – Mathematical Reasoning
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16.Statistics
- Mean, Median, Mode, Range, Mean Deviation Solution of Ex-15.1
- Mean Deviation about Mean & Median, Ex-15.2, Mean and Variance, Standard deviation
- Ex-15.2 , Variance and Standard deviation
- Ex-15.3, Analysis of frequency distribution, comparison of two frequency distribution with same mean
- Chapter Notes – Statistics
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17.Probability
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18.Binary Number
NCERT Solutions – Sets, Relation and Functions
Exercise 1.1
1. Which of the following are sets? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-Cricket batmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in the chapter.
(ix) A collection of most dangerous animals of the world.
Ans. (i) The collection of all months of a year beginning with J is {January, June, July}, which is well defined and hence it forms a set.
(ii) The collection of most talented writers of India is not well defined because opinions about ‘most talented writers’ vary from person to person and hence it does not form a set.
(iii) A team of eleven best-cricket batmen of the world us not well defined because opinion about ‘best-cricket batsmen’ vary from person to person and hence it does not form a set.
(iv) The collection of all boys in your class is well defined and hence it forms a set.
(v) The collection of all natural numbers less than 100 is {1, 2, 3, ……., 99} which is well defined and hence it forms a set.
(vi) A collection of novels written by the writer Munshi Prem Chand is well defined and hence it forms a set.
(vii) The collection al all even integers is 
which is well defined and hence it forms a set.
(viii) The collection of questions in this chapter is well defined and hence it forms a set.
(ix) A collection of most dangerous animals of the world is not well defined because opinion about ‘most dangerous animals’ vary from person to person and hence it does not form a set.
2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol 
or 
on the blank space:
(i) 5 _______ A
(ii) 8 _______ A
(iii) 0 _______ A
(iv) 4 _______ A
(v) 2 _______ A
(vi) 10 _______ A
Ans. Given: A = {1, 2, 3, 4, 5, 6}
(i) 5 is an element of set A
5 A
(ii) 8 is not an element of set A
8 A
(iii) 0 is not an element of set A
0 A
(iv) 4 is an element of set A
4 A
(v) 2 is an element of set A
2 A
(vi) 10 is not an element of set A
10 A
3. Write the following sets in roster form:
(i) A = {
is an integer and 
}
(ii) B = { is a natural number less than 6}
(iii) C = { is a two-digit natural number such that the sum of its digits is 8}
(iv) D = { is a prime number which is divisor of 60}
(v) E = The set of all letters in the word TRIGONOMETRY
(vi) F = The set of all letters in the word BETTER
Ans. (i) A = { is an integer and } A = 
(ii) B = { is a natural number less than 6} B = {1, 2, 3, 4 5}
(iii) C = { is a two-digit natural number such that the sum of its digits is 8}
C = {17, 26, 35, 44, 53, 62, 71, 80}
(iv) D = { is a prime number which is divisor of 60}
D = {2, 3, 5}
(v) E = The set of all letters in the word TRIGONOMETRY
E = {T, R, I, G, O, N, O, M, E, T, R, Y}
(vi) F = The set of all letters in the word BETTER
F = {B, E, T, R}
4. Write the following sets in the set-builder form:
(i) {3, 6, 9, 12}
(ii) {2, 4, 8, 16, 32}
(iii) {5, 25, 125, 625}
(iv) {2, 4, 6, ……}
(v) {1, 4, 9, ………, 100}
Ans. (i) Let A = {3, 6, 9, 12}.
Here all objects of the set are natural numbers that are multiples of 3.
A = 
(ii) Let B = {2, 4, 8, 16, 32}
Here all objects of the set are natural numbers that are power of 2.
B = 
(iii) Let C = {5, 25, 125, 625}
Here all objects of the set are natural numbers that are power of 5.
C = 
(iv) Let D = {2, 4, 6, ……}
Here all objects of the set are even natural numbers.
D = 
(v) Let E = {1, 4, 9, ………, 100}
Here all objects of the set are perfect square.
D = 
5. List all the elements of the following sets:
(i) A = { is an odd natural number}
(ii) B = { is an integer, 
}
(iii) C = { is an integer, 
}
(iv) D = { is a letter in the word “LOYAL”}
(v) E = { is a month of a year not having 31 days}
(vi) F = { is a consonant in the English alphabet which precedes K}
Ans. (i) A = { is an odd natural number}
A = {1, 3, 5, 7, ……..}
(ii) B = { is an integer, }
B = {0, 1, 2, 3, 4}
(iii) C = { is an integer, }
C = 
(iv) D = { is a letter in the word “LOYAL”}
D = {L, O, Y, A}
(v) E = { is a month of a year not having 31 days}
E = {February, April, June, September, November}
(vi) F = { is a consonant in the English alphabet which precedes K}
F = {B, C, D, F, G, H, J}
6. Match each of the set 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) { is a prime number and a divisor of 6}
(ii) {2, 3}
(b) { is an odd natural number less than 10}
(iii) {M, A, T, H, E, I, C, S}
(c) { is a natural number and divisor of 6}
(iv) {!, 3, 5, 7, 9}
(d) { is a letter of word “MATHEMATICS”}
Ans. The sets which are in set-builder form can be written as
(a) { is a prime number and a divisor of 6} = {2, 3}
(b) { is an odd natural number less than 10} = {1, 3, 5, 7, 9}
(c) { is a natural number and divisor of 6} = {1, 2, 3, 6}
(d) { is a letter of word “MATHEMATICS”} = {M, A, T, H, E, I, C, S}
Hence the correct matching is:
(i) 
(c)
(ii) (a)
(iii) (d)
(iv) (b)
Exercise 1.2
1. Which of the following are examples of the null set:
(i) Set of odd natural numbers divisible by 2.
(ii) Set of even prime numbers.
(iii) {
is a natural number, 
and 
}
(iv) {
is a point common to any two parallel lines}
Ans. (i) Set of odd natural numbers divisible by 2 is an empty set because odd natural numbers are not divisible by 2.
(ii) Set of even prime numbers is {2} which is not empty set.
(iii) { is a natural number, and } is an empty set because there is no natural number which satisfies simultaneously and .
(iv) { is a point common to any two parallel lines} is an empty set because two parallel lines do not have a common point.
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.
Ans. (i) The set of months of a year is finite set because there are 12 months in a year.
(ii) {1, 2, 3, ………..} is an infinite set because there are infinite elements in the set.
(iii) {1, 2, 3, ………….., 99, 100} is a finite set because the set contains finite number of elements.
(iv) The set of positive integers greater than 100 is an infinite set because there are infinite number of positive integers greater than 100.
(v) The set of prime numbers less than 99 is a finite set because the set contains finite number of elements.
3. State whether each of the following sets is finite or infinite:
(i) The set of lines which are parallel to the 
axis.
(ii) The set of letters in the English alphabet.
(iii) The set of numbers which are multiple of 5.
(iv) The set of animals living on the earth.
(v) The set of circles passing through the origin (0, 0).
Ans. (i) The set of lines which are parallel to the axis is an infinite set because we can draw infinite number of lines parallel to axis.
(ii) The set of letters in the English alphabet is a finite set because there are 26 letters in the English alphabet.
(iii) The set of numbers which are multiple of 5 is an infinite set because there are infinite multiples of 5.
(iv) The set of animals living on the earth is a finite set because the number of animals living on the earth is every large but finite.
(v) The set of circles passing through the origin (0, 0) is an infinite set because we can draw infinite number of circles through origin in different radii.
4. In the following, state whether A = B or not:
(i) A = 
B = 
(ii) A = {4, 8, 12, 16} B = {8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10} B = { is a positive even integer and 
}
(iv) A = { is a multiple of 10} B = {10, 15, 20, 25, 30, ….}
Ans. (i) A = and B = are equal sets because order of elements does not change a set. Therefore, A = B =
(ii) A = {4, 8, 12, 16} and B = {8, 4, 16, 18} are not equal sets because 12 
A, 12 
B and 18 B, 19 A
(iii) A = {2, 4, 6, 8, 10} and B = { is a positive even integer and } which can be written in roster form as B = {2, 4, 6, 8, 10} are equal sets.
(iv) A = { is a multiple of 10} can be written in roster form as A = {10, 20, 30, 40, …..} and B = {10, 15, 20, 25, 30, ….} are not equal sets because 15 B, 15 A.
5. Are the following pairs of sets equal? Give reason.
(i) A = {2, 3} and B = { is a solution of 
}
(ii) A = : is a letter in the word FOLLOW}
B = { is a letter in the word WOLF}
Ans. (i) A = {2, 3} and B
= { is a solution of }
Here
B = 
Therefore, A and B are not equal sets.
(ii) A = : is a letter in the word FOLLOW} = {F, O, L, W}
B = { is a letter in the word WOLF} = {W, O, L, F}
Therefore, A = B = {F, O, L, W}
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 = 
F = 
G = 
H = {0, 1}
Ans. From the given sets, Set B and D have same elements and also sets E and G have same element.
Exercise 1.3
1. Make correct statements by filling in the symbols 
or 
in the blank spaces:
(i) {2, 3, 4} _______ {1, 2, 3, 4, 5}
(ii) 
(iii) {
is a student of class XI of your school} _______ { student of your school}
(iv) { is a circle in the plane} _______ { is a circle in the same plane with 1 unit}
(v) { is a triangle in plane} _______ {is a rectangle in the same plane}
(vi) { is an equilateral triangle in a plane} _______ { is a rectangle in the same plane}
(vii) { is an even natural number} _______ { is an integer}
Ans. (i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
2. Examine whether the following statements are true or false:
(i) 
(ii) 
{ is a vowel in the English alphabet}
(iii) 
(iv) 
(v) 
(vi) { is an even natural number less than 6}{ is a natural number which divide 36}
Ans. (i) Let A = 
and B = 
Here, every element of set A is an element of set B.
A B
Therefore, statement is false.
(ii) Let A = 
and B
= { is a vowel in the English alphabet}
= 
Here, every element of set A is an element of set B.
A B
Therefore, statement is true.
(iii) Let A = {1, 2, 3} and B = {1, 3, 5}
Here, 2 
A but 2 
B
A B
Therefore, statement is false.
(iv) Let A = 
and B = 
Here, every element of set A is an element of set B.
A B
Therefore, statement is true.
(v) Let A = and B =
Here, B
Therefore, statement is false.
(vi) Let A = { is an even natural number less than 6}
= {2, 4}
And B = }{ is a natural number which divide 36}
= {1, 2, 3, 4, 6, 12, 18, 36]
Here, every element of set A is an element of set B.
A B
Therefore, statement is true.
3. Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why:
(i) {3, 4} A
(ii) {3, 4} A
(iii) {{3, 4}} A
(iv) 1 A
(v) 1 A
(vi) {1, 2, 5} A
(vii) {1, 2, 5} A
(viii) {1, 2, 3} A
(ix) 
A
(x) 
A
Ans. (i) {3, 4} is a member of set A.
{3, 4} A
Therefore, {3, 4} A is incorrect.
(ii) {3, 4} is a member of set A. Therefore, {3, 4} A is incorrect.
(iii) {3, 4} is a member of set A.
{{3, 4}} is a set.
Therefore, {{3, 4}} A is incorrect
(iv) 1 is a member of set A. Therefore 1 A is correct.
(v) 1 is not a set, it is a member of set A. Therefore, 1 A is incorrect.
(vi) 1, 2, 5 are the members of set A.
{1, 2, 5} is a subset of set A.
Therefore, {1, 2, 5} A is correct.
(vii) 1, 2, 5 are the members of set A.
{1, 2, 5} is a subset of set A.
Therefore, {1, 2, 5} A is incorrect.
(viii) 3 is not a member of set A.
{1, 2, 3} is not a subset of set A.
Therefore, {1, 2, 3} A is incorrect.
(ix) 
is not a member of set A. Therefore, A is correct.
(x) is not a member of set A. Therefore, A is incorrect.
4. Write down all the subsets of the following sets:
(i)
(ii)
(iii) {1, 2, 3}
(iv)
Ans. (i) Number of elements in given set = 1.
Number of subsets of given set = 
= 2
Therefore, Subsets of given set are 
(ii) Number of elements in given set = 2
Number of subsets of given set = 
= 4
Therefore, Subsets of given set are
(iii) Number of elements in given set = 3
Number of subsets of given set = 
= 8
Therefore, Subsets of given set are
(iv) Number of elements in given set = 0
Number of subsets of given set = 
= 1
Therefore, Subsets of given set are 
5. How many elements has P(A), if A = ?
Ans. Number of elements in set A = 0
Number of subsets of given set = = 1
Therefore, number of elements of P (A) is 1.
6. Write the following as intervals:
(i) { R, 
}
(ii) { R, 
}
(iii) { R, 
}
(iv) { R, 
}
Ans. (i) Let A = { R, }
It can be written in the form of interval as 
(ii) Let A = { R, }
It can be written in the form of interval as 
(iii) Let A = { R, }
It can be written in the form of interval as 
(iv) Let A = { R, }
It can be written in the form of interval as 
7. Write the following intervals in set-builder form:
(i) 
(ii) [6, 12]
(iii) (6, 12]
(iv) 
Ans. (i) { R, 
}
(ii) { R, 
}
(iii) { R, 
}
(iv) { R, 
}
8. What universal set(s) would you propose for each of the following:
(i) The set of right triangles
(ii) The set of isosceles triangles
Ans. (i) Right triangle is a type of triangle. Therefore, the set of triangles contain all types of triangles.
U = { is a triangle in plane}
(ii) Isosceles triangle is a type of triangle. Therefore, the set of triangles contain all types of triangles.
U = { is a triangle in plane}
9. Given the set A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set(s) for all the three sets A, B and C:
(i) {0, 1, 2, 3, 4, 5, 6}
(ii)
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv) {1, 2, 3, 4, 5, 6, 7, 8}
Ans. (i) {0, 1, 2, 3, 4, 5, 6} is not a universal set for A, B, C because 8 C but 8 is not a member of {0, 1, 2, 3, 4, 5, 6}.
(ii) is a set which contains no element. therefore, it is not a universal set for A, B, C.
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is a universal set for A, B, C because all members of A, B, C are present in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
(iv) {1, 2, 3, 4, 5, 6, 7, 8} is not a universal set for A, B, C because 0 C but 0 is not a member of {1, 2, 3, 4, 5, 6, 7, 8}.
Exercise 1.4
1. Find the union of each of the following pairs of sets:
(i) X = {1, 3, 5} and Y = {1, 2, 3}
(ii) A = 
and B = 
(iii) A = {
is a natural number and multiple of 3} and B = { is a natural number less than 6}
(iv) A = { is a natural number and 
} and B = { is a natural number and 
}
(v) A = {1, 2, 3} and B = 
Ans. (i) X 
Y = {1, 2, 3, 5}
(ii) A B = 
(iii) A B = {1, 2, 3, 4, 5, 6, 9, 12, 15, ………..}
(iv) A B = {2, 3, 4, 5, 6, 7, 8, 9}
(v) A B = {1, 2, 3}
2. Let A = 
and B = . Is A 
B? What is A B?
Ans. Given: A = and B = .
Here all elements of set A are present in set B.
A B and A B = = B
3. If A and B are two sets such that A B, then what is A B?
Ans. Given: A and B are two sets such that A B
Taking A = {1, 2} and B = {1, 2, 3}, then A B = {1, 2, 3} = B
4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find:
(i) A B
(ii) A C
(iii) B C
(iv) B D
(v) A B C
(vi) A B D
(vii) B C D
Ans. Given: A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C
= {5, 6, 7, 8} and D = {7, 8, 9, 10}
(i) A B = {1, 2, 3, 4} {3, 4, 5, 6}
= {1, 2, 3, 4, 5, 6}
(ii) A C = {1, 2, 3, 4} {5, 6, 7, 8}
= {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B C = {3, 4, 5, 6} {5, 6, 7, 8}
= {3, 4, 5, 6, 7, 8}
(iv) B D = {3, 4, 5, 6} {7, 8, 9, 10}
= {3, 4, 5, 6, 7, 8, 9, 10}
(v) A B C = {1, 2, 3, 4} {3, 4, 5, 6 {5, 6, 7, 8}
= {1, 2, 3, 4, 5, 6, 7, 8}
(vi) A B D = {1, 2, 3, 4} {3, 4, 5, 6} {7, 8, 9, 10}
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B C D = {3, 4, 5, 6} {5, 6, 7, 8} {7, 8, 9, 10}
= {3, 4, 5, 6, 7, 8, 9, 10}
5. Find the intersections of each pair of sets of question 1 above.
Ans. (i) X 
Y = {1, 3}
(ii) A B = 
(iii) A B = {3, 6, 9, 12, ………..}
(iv) A B =
(v) A B =
6. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find:
(i) A B
(ii) B C
(iii) A C D
(iv) A C
(v) B D
(vi) A (B C)
(vii) A D
(viii) A (B D)
(ix) (A B) (B C)
(x) (A D) (B C)
Ans. Given: A = {3, 5, 7, 9, 11},
B = {7, 9, 11, 13},
C = {11, 13, 15} and D = {15, 17}
(i) A B = {3, 5, 7, 9, 11} {7, 9, 11, 13}
= {7, 9, 11}
(ii) B C = {7, 9, 11, 13} {11, 13, 15}
= {11, 13}
(iii) A C D
= {3, 5, 7, 9, 11} {11, 13, 15} {15, 17} =
(iv) A C = {3, 5, 7, 9, 11} {11, 13, 15}
= {11}
(v) B D = {7, 9, 11, 13} {15, 17} =
(vi) A (B C)
= {3, 5, 7, 9, 11} ({7, 9, 11, 13} {11, 13, 15})
= {3, 5, 7, 9, 11} {7, 9, 11, 13, 15, 17} = {7, 9, 11}
(vii) A C = {3, 5, 7, 9, 11} {15, 17} =
(viii) A (B D)
= {3, 5, 7, 9, 11} ({7, 9, 11, 13} {15, 17})
= {3, 5, 7, 9, 11} {7, 9, 11, 13, 15, 17}
= {7, 9, 11}
(ix) (A B) (B C)
= ({3, 5, 7, 9, 11} {7, 9, 11, 13}) ({7, 9, 11, 13} {11, 13, 15})
= {7, 9, 11} {7, 9, 11, 13, 15} = {7, 9, 11}
(x) (A D) (B C)
= ({3, 5, 7, 9, 11} {15, 17}) ({7, 9, 11, 13} {11, 13, 15})
= {3, 5, 7, 9, 11, 15, 17} {7, 9, 11, 13, 15}
= {7, 9, 11, 15}
7. If A = { is a natural number}, B = { is an even natural number}, C = { is an odd natural number} and D = { is a prime number}, find:
(i) A B
(ii) A C
(iii) A D
(iv) B C
(v) B D
(vi) C D
Ans. (i) A B = { is a natural number} { is an even natural number} = B
(ii) A C = { is a natural number} { is an odd natural number} = C
(iii) A D = { is a natural number} { is a prime number} = D
(iv) B C = { is an even natural number} { is an odd natural number} =
(v) B D = B C = { is an even natural number} { is a prime number} = {2}
(vi) C D = { is an odd natural number} { is a prime number}
= { is an odd prime number}
8. Which of the following pair of sets are disjoint:
(i) {1, 2, 3, 4} and { is a natural number and 
}
(ii) and 
(iii) { is an even integer} and { is an odd integer}
Ans. (i) Let A = {1, 2, 3, 4} and B = { is a natural number and } = {4, 5, 6}
A B = {4}
Therefore, A and B are not disjoint.
(ii) Let A = and B =
A B =
Therefore, A and B are disjoint.
(iii) Let A = { is an even integer} and B = { is an odd integer}
A B =
Therefore, A and B are disjoint.
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
Ans. Given: 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};
(i) A – B = {3, 6, 9, 12, 15, 18, 21} – {4, 8, 12, 16, 20}
= {3, 6, 9, 15, 18, 21}
(ii) A – B = {3, 6, 9, 12, 15, 18, 21} – {2, 4, 6, 8, 10, 12 , 14, 16}
= {3, 9, 15, 18, 21}
(iii) A – B = {3, 6, 9, 12, 15, 18, 21} – {5, 10, 15, 20}
= {3, 6, 9, 15, 18, 21}
(iv) B – A = {4, 8, 12, 16, 20} – {3, 6, 9, 12, 15, 18, 21}
= {4, 8, 16, 20}
(v) C – A = {2, 4, 6, 8, 10, 12 , 14, 16} – {3, 6, 9, 12, 15, 18, 21}
= {2, 4, 8, 14, 16}
(vi) D – A = {5, 10, 15, 20} – {3, 6, 9, 12, 15, 18, 21}
= {5, 10, 20}
(vii) B – C = {4, 8, 12, 16, 20} – {2, 4, 6, 8, 10, 12 , 14, 16} = {20}
(viii) B – D = {4, 8, 12, 16, 20} – {5, 10, 15, 20}
= {4, 8, 12, 16}
(ix) C – B = {2, 4, 6, 8, 10, 12 , 14, 16} – {4, 8, 12, 16, 20}
= {2, 6, 10, 14}
(x) D – B = {5, 10, 15, 20} – {4, 8, 12, 16, 20}
= {5, 10, 15}
(xi) C – D = {2, 4, 6, 8, 10, 12 , 14, 16} – {5, 10, 15, 20}
= {2, 4, 6, 8, 12, 14, 16}
(xii) D – C = {5, 10, 15, 20} – {2, 4, 6, 8, 10, 12 , 14, 16}
= {5, 15, 20}
10. If X = 
and Y = 
find:
(i) X – Y
(ii) Y – X
(iii) X Y
Ans. Given: X = and Y = 
(i) X – Y = 
= 
(ii) Y – X = 
= 
(iii) X Y = 
= 
11. If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?
Ans. We know that set of real numbers contain rational and irrational numbers.
Therefore, R – Q = set of irrational numbers.
12. State whether each of the following statements is true or false. Justify your answer.
(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.
(ii) and are disjoint sets.
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.
Ans. (i) Let A = {2, 3, 4, 5} and B = {3, 6}
A B = {3}
A and B are not disjoint. Therefore, statement is false.
(ii) Let A = and B =
A B =
A and B are not disjoint. Therefore, statement is false.
(iii) Let A = {2, 6, 10, 14} and B = {3, 7, 11, 15}
A B =
A and B are disjoint. Therefore, statement is true.
(iv) Let A = {2, 6, 10} and B = {3, 7, 11}
A B =
A and B are disjoint. Therefore, statement is true.
Exercise 1.5
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 
C)’
(iv) (A B)’
(v) (A’)’
(vi) (B – C)’
Ans. Given: 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 = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4}
= {5, 6, 7, 8, 9}
(ii) B’ = U – B = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}
= {1, 3, 5, 7, 9}
(iii) (A C)’ = U – (A C)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({1, 2, 3, 4} {3, 4, 5, 6})
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 5, 6} = {7, 8, 9}
(iv) (A B)’ = U – (A B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({1, 2, 3, 4} {2, 4, 6, 8})
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 6, 8} = {5, 7, 9}
(v) (A’)’ = U – A’ = U – (U – A)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4})
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {5, 6, 7, 8, 9}
= {1, 2, 3, 4} = A
(vi) (B – C)’ = U – (B – C)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({2, 4, 6, 8} – {3, 4, 5, 6})
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 8}
= {1, 3, 4, 5, 6, 7, 9}
2. If U = 
find the complement of the following sets:
(i) A = 
(ii) B = 
(iii) C = 
(iv) D = 
Ans. Given: U = 
(i) A’ = U – A
= 
(ii) B’ = U – B
= 
(iii) C’ = U – C
= 
(iv) D’ = U – D
= 
3. Taking the set of natural numbers as the universal set, write down the complement of the following set:
(i) {
is an even natural number}
(ii) { is an odd natural number}
(iii) { is a positive multiple of 3}
(iv) { is a prime number}
(v) { is a natural number divisible by 3 and 5}
(vi) { is a perfect square}
(vii) { is a perfect cube}
(viii) { + 5 = 8}
(ix) {
+5=9}
(x) 
(xi) { 
N and 
}
Ans. Given: U = 
(i) Let A = { is an even natural number}
A’ = U – A = – { is an even natural number}
= { is an odd natural number}
(ii) Let A = { is an odd natural number}
A’ = U – A = – { is an odd natural number}
= { is an even natural number}
(iii) Let A = { is a positive multiple of 3}
A’ = U – A = – { is a positive multiple of 3}
= {
is not a positive multiple of 3}
(iv) Let A = { is a prime number}
A’ = U – A = – { is a prime number}
= { is not a prime number}
(v) Let A = { is a natural number divisible by 3 and 5}
A’ = U – A = – { is a natural number divisible by 15}
= { is not divisible by 15}
(vi) Let A = { is a perfect square}
A’ = U – A = – { is a perfect square}
= { is not a perfect square}
(vii) Let A = { is a perfect cube}
A’ = U – A = – { is a perfect cube}
= { is not a perfect cube}
(viii) Let A = { + 5 = 8} = {3}
A’ = U – A = – {3}
= { 
3}
(ix) Let A = { +5=9} = {2}
A’ = U – A = – {2}
= { 2}
(x) Let A = = {7, 8, 9, 10, ………}
A’ = U – A = – {7, 8, 9, 10, ………}
={1, 2, 3, 4, 5, 6} = { 
7}
(xi) Let A = { N and } = {5, 6, 7, 8, ………..}
A’ = U – A = – {5, 6, 7, 8, ………..}
= {1, 2, 3, 4}
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) 
(ii) 
Ans. Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
A = {2, 4, 6, 8} and B = {2, 3, 5, 7}
(i) L.H.S. = 
= U – (A B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({2, 4, 6, 8} {2, 3, 5, 7})
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 4, 5, 6, 7, 8} = {1, 9}
R.H.S. = 
= (U – A) 
(U – B)
= ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}) ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 5, 7})
= {1, 3, 5, 7, 9} {1, 4, 6, 8, 9} = {1, 9}
L.H.S. = R. H. S.
(ii) L.H.S. = 
= U – (A B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – ({2, 4, 6, 8} {2, 3, 5, 7})
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2}
= {1, 3, 4, 5, 6, 7, 8, 9}
R.H.S. = 
= (U – A) (U – B)
= ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}) ({1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 3, 5, 7})
= {1, 3, 5, 7, 9} {1, 4, 6, 8, 9}
= {1, 3, 4, 5, 6, 7, 8, 9}
L.H.S. = R. H. S.
5. Draw appropriate Venn diagrams for each of the following:
(i)
(ii)
(iii)
(iv)
Ans. (i) In the diagrams, shaded portion represents
(ii) In the diagrams, shaded portion represents
(iii) In the diagrams, shaded portion represents
(iv) In the diagrams, shaded portion represents
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 
what is A’?
Ans. Given: U = { is a triangle}
A = { is a triangle and has at least one angle different from 
}
A’ = U – A = { is a triangle and has all angles equal to }
= Set of all equilateral triangles
7. Fill in the blanks to make each of the following a true statement:
(i) 
(ii) 
(iii) 
(iv) 
Ans. (i) 
(ii) 
(iii) 
(iv) 
Exercise 1.6
1. If X and Y are two sets such that 
and 
find 
Ans. Given: 
and 
2. If X and Y are two sets such that 
has 18, X has 8 elements and Y has 15 elements; how many elements has 
?
Ans. Given: 
and 
3. In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?
Ans. Let H be the set of people speaking Hindi and E be the set of people speaking English.
and 
4. If S and T are two sets such that S has 21 elements T has 32 elements and 
has 11 elements, how many elements does 
have?
Ans. Given: 
and 
5. If X and Y are two sets such that X has 40 elements, has 60 elements and has 10 elements, how many elements does Y have?
Ans. Given: 
and 
6. In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?
Ans. Given: 
and 
7. In a group of 65 people. 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
Ans. Let C be the set of people who like cricket and T be the set of people who like tennis.
Then 
and 
Therefore, number of people who like tennis are 35.
Now number of people who like tennis only and not cricket = 
= 35 – 10 = 25
8. In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?
Ans. Let F be the set of people who speak French and S be the set of people who speak Spanish.
Then 
and 
Therefore, Number of people who speak at least one of these two languages are 60.
Miscellaneous Exercise
1. Decide among the following sets, which sets are subsets of each another:
A = {
R and 
satisfies 
}, B = {2, 4, 6}, C = {2, 4, ,6 , 8…….}, D = {6}
Ans. Given: A = { R and satisfies }
= { R and satisfies 
} = {2, 6}
B = {2, 4, 6}, C = {2, 4, 6, 8…….}, D = {6}
A 
B, A C, B C, D A, D B and D C
2. 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.
(i) If 
A and A 
B then B
(ii) If A B and B C then A C
(iii) If A B and B C then A C
(iv) If A 
B and B C then A C
(v) If A and A B then B
(vi) If A B and 
C then A
Ans. (i) The statement is false.
Let A = {1} and B = {{1}, 2}
Then 1 A and A B but 1 
B
(ii) The statement is false.
Let A = {1} and B = {1, 2}, C {{1, 2}, 3}
Then A B and B C but A C
(iii) The statement is true.
Let A 
B (
A B)
C ( B C)
A C A C
(iv) The statement is false.
Let A = {1, 2}, B = {2, 3}, C {1, 2, 5}
Then A B and B C but A C
(v) The statement is false.
Let A = {1, 2} and B = {2, 3, 4, 5}
Then 1 A and A B but 1 B
(vi) The statement is true.
Let A B ( A B)
Now, B A
3. Let A, B and C be the sets such that A 
B = A C and A 
B = A C, then show that B = C.
Ans. Since, A = A (A B) and A = A (A B)
Now, it is given that A B = A C and A B = A C
B = B (B A) = B (A B) = B (A C)
= (B A) (B C) = (A B) (B C) = (A C) (B C)
= (C A) (C B) = C (A B) = C (A C)
= C (C A) = C
B = C
4. Show that the following four conditions are equivalent:
(i) A B
(ii) A – B = 
(iii) A B = B
(iv) A B = A
Ans. (i) (ii) A – B = { A and B}
Since A B, Therefore A – B =
(ii) (iii) A – B =
A B A B = B
(iii) (iv) A B = B
A B A B = A
(iv) (i) A B = A A B
Therefore, (i) 
(ii) (iii) (iv)
5. Show that if A B, then C – B C – A.
Ans. Let C – B
C and B
C and A [ A B]
C – A C – B C – A
6. Assume that P(A) = P(B), show that A = B
Ans. Let A 
P (A)
P (B) B
A B………(i)
Let B P (B)
P (A) A
B A………(ii)
From eq. (i) and (ii), we have A = B
7. Is it true that for any set A and B, P(A) P(B) = P(A B)? Justify your answer.
Ans. No, it is not true.
Taking A = {1, 2} and B = {2, 3}
Then A B = {1, 2, 3}
P (A) = 
and P (B) = 
P (A) P (B) = 
……….(i)
And P (A B) = 
……….(ii)
From eq. (i) and (ii), P (A) P (B) 
P (A B)
8. Show that for any sets A and B, A = (A B) (A – B) and A (B – A) = (A B)
Ans. Since (A B) (A – B) = (A B) (A B’)
(A B) (A – B) = A B B’ = A U = A
Therefore, A = (A B) (A – B)
Also A (B – A) = A (B A’) = (A B) (A A’) = (A B) U = A B
Therefore, A (B – A) = A B
9. Using properties of sets, show that:
(i) A (A B) = A
(ii) A (A B) = A
Ans. (i) If A B, then A B = B
Also A B A
A (A B) = A
(ii) If A B, then A B = A
Also A A B
A (A B) = A
10. Show that A B = A C need not imply B = C.
Ans. Let A = {1, 2, 3, 4}, B = {2, 3, 4, 5, 6} and C = {2, 3, 4, 9, 10}
A B = {2, 3, 4}
And A C = {2, 3, 4}
Therefore, we have A B = A C
But B C
11. Let A and B sets. If A X = B X = and A X = B X for some set X. Show that A = B.
[Hint: A = A (A X), B = B (B X) and use Distributive law]
Ans. Given: A X = B X for some set X.
A (A X) = A (B X)
A = (A B) (A X)
A = (A B)
A = (A B)
A B……….(i)
Also A X = B X
B (A X) = B (B X)
(B A) (B X) = B
(B A) = B
B A = BB A ……….(ii)
From eq. (i) and (ii), we have A = B
12. Find sets A, B and C such that A B, B C and A C are non-empty sets and A B C = 
Ans. Let A = {1, 2}, B = {1, 4} and C = {2, 4}
A B = {1} B C = {4}
And A C = {2}
But A B C =
13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee.
Ans. Let T be the set of students who like tea and C be the set of students who like coffee.
and 
Number of students taking either tea or coffee = 275
Number of students taking neither tea nor coffee = 600 – 275 = 325
14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?
Ans. Let H be the set of students who know Hindi and E be the set of students who know English.
and 
15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 real all three newspapers. Find:
(i) the number of people who read at least one of the newspaper.
(ii) the number of people who read exactly one newspaper.
Ans. Given: 
…..(i)
…..(ii)
…..(iii)
…..(iv)
…..(v)
…..(vi)
…..(vii)
…..(viii)
Putting value of 
in eq. (vii), 
Putting value of in eq. (vi), 
Putting value of in eq. (v), 
Putting value of 
in eq. (iv), 
Putting value of in eq. (iii), 
Putting value of 
in eq. (ii), 
(i) Number of people who read at least one of the three newspapers
= 
= 8 + 8 + 3 + 6 + 12 + 5 + 10 = 52
(ii) Number of people who read exactly one newspaper
= 
16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B,12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.
Ans. Given: 
…..(i)
…..(ii)
…..(iii)
…..(iv)
…..(v)
…..(vi)
…..(vii)
Putting value of in eq. (iv), 
Putting value of in eq. (v), 
Putting value of in eq. (vi), 
Putting value of in eq. (iii), 
Number of people who like product C only = 11.