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      Class 11 Maths

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      • Class 11
      • Class 11 Maths
      CoursesClass 11MathsClass 11 Maths
      • 1.Basic Mathematics
        11
        • Lecture1.1
          Sets 55 min
        • Lecture1.2
          Basic Inequalities 43 min
        • Lecture1.3
          Wavy Curve Method 38 min
        • Lecture1.4
          Relations 36 min
        • Lecture1.5
          Introduction to Functions 32 min
        • Lecture1.6
          Domain of a Function 44 min
        • Lecture1.7
          Range of Functions 43 min
        • Lecture1.8
          Basic Functions-Constant Function, Identity Function, Modulus Function 52 min
        • Lecture1.9
          Basic Functions-Greatest Integer Function, Fractional Part Function 40 min
        • Lecture1.10
          Basic Functions- Signum Function, Exponential Function, Logarithmic Function 23 min
        • Lecture1.11
          Basic Mathematics – NCERT Solutions
      • 2.Trigonometry
        10
        • Lecture2.1
          Measurement of Angles 31 min
        • Lecture2.2
          Trigonometric Ratios of Allied Angles 33 min
        • Lecture2.3
          Sum and Difference Formulas 56 min
        • Lecture2.4
          Transforming Product into Sum and Difference of Angles 52 min
        • Lecture2.5
          C-D Formulas and Double Angle Formulas 40 min
        • Lecture2.6
          Triples Angle Formulas and Cosine Series 33 min
        • Lecture2.7
          Conditional Identities 45 min
        • Lecture2.8
          Trigonometric Equations 46 min
        • Lecture2.9
          Solution of Triangles 27 min
        • Lecture2.10
          Graphs of Trigonometric Functions 27 min
      • 3.Mathematical Induction
        4
        • Lecture3.1
          Introduction to PMI 25 min
        • Lecture3.2
          NCERT Solution of EX- 4.1 15 min
        • Lecture3.3
          NCERT Solution of EX- 4.1 22 min
        • Lecture3.4
          NCERT Solution of EX- 4.1 13 min
      • 4.Quadratic Equations
        4
        • Lecture4.1
          Introduction to Quadratic Equations, Sum and Product of Roots, Discriminant 39 min
        • Lecture4.2
          Formation of QE, QE as an Identity, Graphs of QE 58 min
        • Lecture4.3
          Condition for Common Root, Cubic Equation, Transformation of Equation 45 min
        • Lecture4.4
          Location of Roots 37 min
      • 5.Complex Number
        5
        • Lecture5.1
          Iota and its Integral Powers, Equalities and Inequalities of a Complex Number 35 min
        • Lecture5.2
          Algebraic Operations, Conjugate of a Complex Number 46 min
        • Lecture5.3
          Modulus of a Complex Number 42 min
        • Lecture5.4
          Argument, Square Root, Logarithm 42 min
        • Lecture5.5
          De Moivre’s Theorem, Nth Roots and Cube roots of Unity 48 min
      • 6.Binomial Theorem
        4
        • Lecture6.1
          Factorial 24 min
        • Lecture6.2
          Binomial Theorem 33 min
        • Lecture6.3
          Problems, Middle Term, Greatest Binomial Coefficient 27 min
        • Lecture6.4
          Numerically Greatest Term, Properties of Binomial Coefficients, Binomial Theorem for Rational Index 24 min
      • 7.Permutation and Combination
        4
        • Lecture7.1
          Fundamental Principles of Counting 27 min
        • Lecture7.2
          Linear Permutation, Arrangement of like objects, Selection of Objects, Misc Problems 39 min
        • Lecture7.3
          Grouping Method, Gap Method 19 min
        • Lecture7.4
          Formation of groups, Beggar’s Method, Circular Permutation 48 min
      • 8.Sequences and Series
        7
        • Lecture8.1
          Arithmetic Progression 29 min
        • Lecture8.2
          Sum of an A.P. 44 min
        • Lecture8.3
          Geometric Progression 31 min
        • Lecture8.4
          Infinite G.P. 24 min
        • Lecture8.5
          Harmonic Progression 20 min
        • Lecture8.6
          AM, GM, HM and Inequalities 51 min
        • Lecture8.7
          Arithmetico-Geometric Progression, Special Series 34 min
      • 9.Straight Lines
        6
        • Lecture9.1
          Basic Concepts, Forms of Straight Lines 34 min
        • Lecture9.2
          Examples 23 min
        • Lecture9.3
          Forms of Straight Lines, Conversion of One Form to Another 39 min
        • Lecture9.4
          General Form, Position of Points, Length of Perpendicular, Distance between Parallel lines 41 min
        • Lecture9.5
          Angle between two lines, Family of Lines, Area of Parallelogram, Concurrent Lines 41 min
        • Lecture9.6
          Image in a line, Shifting of Origin, Angular Bisectors 33 min
      • 10.Circle
        6
        • Lecture10.1
          Introduction and General Equation of Circle 35 min
        • Lecture10.2
          Diametric Form, Intercept made by circle, Position of a point, Parametric form 40 min
        • Lecture10.3
          Position of a line wrt Circle, Condition of Tangency 20 min
        • Lecture10.4
          Equation of Tangent at a point on the circle 27 min
        • Lecture10.5
          Equation of Tangent from External point, Geometry Related to circles, Chord 39 min
        • Lecture10.6
          Common Tangents, Equation of common chord, Power of a point and Radical Axis, Normal to a circle, Orthogonal Circles, Family of circles. 58 min
      • 11.Parabola
        6
        • Lecture11.1
          Introduction to Conic section, Definition, Identifying Conic Section 29 min
        • Lecture11.2
          Introduction to Parabola, Standard Parabolas, Shifted Parabola 34 min
        • Lecture11.3
          Point and Parabola, Parametric Equations, Equation of Chord, Focal Chord and its length, Focal Distance 26 min
        • Lecture11.4
          Line and a Parabola, Equation of Tangent in various forms 31 min
        • Lecture11.5
          Point of intersection of tangents, Common tangents, Normal to a Parabola in Different Forms 31 min
        • Lecture11.6
          Conormal Points, Chord of Contact, Chord bisected at a point, Director Circle 13 min
      • 12.Ellipse
        4
        • Lecture12.1
          Introduction, Standard Ellipses 28 min
        • Lecture12.2
          Shifted Ellipse, Examples, Second Definition of Ellipse 32 min
        • Lecture12.3
          Position of a point w.r.t ellipse, Parametric form, Position of a line, Condition of Tangency, Equation of Tangent in Various Forms 37 min
        • Lecture12.4
          Normal to Ellipse in Point form and Parametric Form, Chord of Contact, Chord Bisected at a point, Director Circle 36 min
      • 13.Hyperbola
        3
        • Lecture13.1
          Introduction, Standard Hyperbolas, Shifted Hyperbola, Conjugate Hyperbola 42 min
        • Lecture13.2
          Parametric Form, Position of Line and Hyperbola, Equation of Tangent in different forms, Director Circle, Chord of Contact, Chord Bisected at a point 24 min
        • Lecture13.3
          Asymptotes, Rectangular Hyperbola, Rotated Rectangular Hyperbola 18 min
      • 14.Introduction to coordinate geometry
        2
        • Lecture14.1
          Distance Formula, Section Formula, Area of Triangle 32 min
        • Lecture14.2
          Centroid, Incenter, Orthocenter, Circumcenter, Locus Problems 37 min
      • 15.Limits and Derivatives
        11
        • Lecture15.1
          Introduction to limits 42 min
        • Lecture15.2
          EX-13.1 16 min
        • Lecture15.3
          Questions based on algebra of limits 41 min
        • Lecture15.4
          limits of a polynomial 12 min
        • Lecture15.5
          Rational Function 37 min
        • Lecture15.6
          Trigono Function 21 min
        • Lecture15.7
          Introduction to Derivatives 37 min
        • Lecture15.8
          Ex-13.2 22 min
        • Lecture15.9
          Algebra of Derivatives 38 min
        • Lecture15.10
          Derivative of polynomial 13 min
        • Lecture15.11
          Trigonometry Function 11 min
      • 16.Mathematical Reasoning
        2
        • Lecture16.1
          Introduction to chapter 24 min
        • Lecture16.2
          Validating statements 24 min
      • 17.Statistics
        4
        • Lecture17.1
          Mean Deviation 27 min
        • Lecture17.2
          Mean and Variance 35 min
        • Lecture17.3
          Standard Deviation 09 min
        • Lecture17.4
          Analysis of frequency distribution 24 min
      • 18.Probability
        2
        • Lecture18.1
          Outcomes and sample space 19 min
        • Lecture18.2
          Probability of an event 38 min

        Basic Mathematics – NCERT Solutions

         

        Page No 4:

        Question 1:

        Which of the following are sets? Justify our answer.

        (i) The collection of all 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 batsmen 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 this Chapter.

        (ix) A collection of most dangerous animals of the world.

        Answer:

        (i) The collection of all months of a year beginning with the letter J is a well-defined collection of objects because one can definitely identify a month that belongs to this collection.

        Hence, this collection is a set.

        (ii) The collection of ten most talented writers of India is not a well-defined collection because the criteria for determining a writer’s talent may vary from person to person.

        Hence, this collection is not a set.

        (iii) A team of eleven best cricket batsmen of the world is not a well-defined collection because the criteria for determining a batsman’s talent may vary from person to person.

        Hence, this collection is not a set.

        (iv) The collection of all boys in your class is a well-defined collection because you can definitely identify a boy who belongs to this collection.

        Hence, this collection is a set.

        (v) The collection of all natural numbers less than 100 is a well-defined collection because one can definitely identify a number that belongs to this collection.

        Hence, this collection is a set.

        (vi) A collection of novels written by the writer Munshi Prem Chand is a well-defined collection because one can definitely identify a book that belongs to this collection.

        Hence, this collection is a set.

        (vii) The collection of all even integers is a well-defined collection because one can definitely identify an even integer that belongs to this collection.

        Hence, this collection is a set.

        (viii) The collection of questions in this chapter is a well-defined collection because one can definitely identify a question that belongs to this chapter.

        Hence, this collection is a set.

        (ix) The collection of most dangerous animals of the world is not a well-defined collection because the criteria for determining the dangerousness of an animal can vary from person to person.

        Hence, this collection is not a set.

        Page No 5:

        Question 2:

        Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈or ∉ in the blank spaces:

        (i) 5…A (ii) 8…A (iii) 0…A

        (iv) 4…A (v) 2…A (vi) 10…A

        Answer:

        (i) 5 ∈ A

        (ii) 8 ∉ A

        (iii) 0 ∉ A

        (iv) 4 ∈ A

        (v) 2 ∈ A

        (vi) 10 ∉ A

        Question 3:

        Write the following sets in roster form:

        (i) A = {x: x is an integer and –3 < x < 7}.

        (ii) B = {x: x is a natural number less than 6}.

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

        (iv) D = {x: x 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.

        Answer:

        (i) A = {x: x is an integer and –3 < x < 7}

        The elements of this set are –2, –1, 0, 1, 2, 3, 4, 5, and 6 only.

        Therefore, the given set can be written in roster form as

        A = {–2, –1, 0, 1, 2, 3, 4, 5, 6}

        (ii) B = {x: x is a natural number less than 6}

        The elements of this set are 1, 2, 3, 4, and 5 only.

        Therefore, the given set can be written in roster form as

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

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

        The elements of this set are 17, 26, 35, 44, 53, 62, 71, and 80 only.

        Therefore, this set can be written in roster form as

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

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

        2 60
        2 30
        3 15
        5

        ∴60 = 2 × 2 × 3 × 5

        The elements of this set are 2, 3, and 5 only.

        Therefore, this set can be written in roster form as D = {2, 3, 5}.

        (v) E = The set of all letters in the word TRIGONOMETRY

        There are 12 letters in the word TRIGONOMETRY, out of which letters T, R, and O are repeated.

        Therefore, this set can be written in roster form as

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

        (vi) F = The set of all letters in the word BETTER

        There are 6 letters in the word BETTER, out of which letters E and T are repeated.

        Therefore, this set can be written in roster form as

        F = {B, E, T, R}

        Question 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}

        Answer:

        (i) {3, 6, 9, 12} = {x: x = 3n, n∈ N and 1 ≤ n ≤ 4}

        (ii) {2, 4, 8, 16, 32}

        It can be seen that 2 = 21, 4 = 22, 8 = 23, 16 = 24, and 32 = 25.

        ∴ {2, 4, 8, 16, 32} = {x: x = 2n, n∈ N and 1 ≤ n ≤ 5}

        (iii) {5, 25, 125, 625}

        It can be seen that 5 = 51, 25 = 52, 125 = 53, and 625 = 54.

        ∴ {5, 25, 125, 625} = {x: x = 5n, n∈N and 1 ≤ n ≤ 4}

        (iv) {2, 4, 6 …}

        It is a set of all even natural numbers.

        ∴ {2, 4, 6 …} = {x: x is an even natural number}

        (v) {1, 4, 9 … 100}

        It can be seen that 1 = 12, 4 = 22, 9 = 32 …100 = 102.

        ∴ {1, 4, 9… 100} = {x: x = n2, n∈N and 1 ≤ n ≤ 10}

        Question 5:

        List all the elements of the following sets:

        (i) A = {x: x is an odd natural number}

        (ii) B = {x: x is an integer,}

        (iii) C = {x: x is an integer,}

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

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

        (vi) F = {x: x is a consonant in the English alphabet which proceeds k}.

        Answer:

        (i) A = {x: x is an odd natural number} = {1, 3, 5, 7, 9 …}

        (ii) B = {x: x is an integer;}

        It can be seen that  and

        ∴ B

        (iii) C = {x: x is an integer;}

        It can be seen that

        (–1)2 = 1 ≤ 4; (–2)2 = 4 ≤ 4; (–3)2 = 9 > 4

        02 = 0 ≤ 4

        12 = 1 ≤ 4

        22 = 4 ≤ 4

        32 = 9 > 4

        ∴C = {–2, –1, 0, 1, 2}

        (iv) D = (x: x is a letter in the word “LOYAL”) = {L, O, Y, A}

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

        = {February, April, June, September, November}

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

        = {b, c, d, f, g, h, j}

        Question 6:

        Match each of the set on the left in the roster form with the same set on the right described in 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 natural number and divisor of 6}
        (iv) {1, 3, 5, 7, 9} (d) {x: x is a letter of the word MATHEMATICS}

         

        Answer:

        (i) All the elements of this set are natural numbers as well as the divisors of 6. Therefore, (i) matches with (c).

        (ii) It can be seen that 2 and 3 are prime numbers. They are also the divisors of 6.

        Therefore, (ii) matches with (a).

        (iii) All the elements of this set are letters of the word MATHEMATICS. Therefore, (iii) matches with (d).

        (iv) All the elements of this set are odd natural numbers less than 10. Therefore, (iv) matches with (b).

        Page No 8:

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

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

        Answer:

        (i) A set of odd natural numbers divisible by 2 is a null set because no odd number is divisible by 2.

        (ii) A set of even prime numbers is not a null set because 2 is an even prime number.

        (iii) {x: x is a natural number, x < 5 and x > 7} is a null set because a number cannot be simultaneously less than 5 and greater than 7.

        (iv) {y: y is a point common to any two parallel lines} is a null set because parallel lines do not intersect. Hence, they have no common point.

        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 set of months of a year is a finite set because it has 12 elements.

        (ii) {1, 2, 3 …} is an infinite set as it has infinite number of natural numbers.

        (iii) {1, 2, 3 …99, 100} is a finite set because the numbers from 1 to 100 are finite in number.

        (iv) The set of positive integers greater than 100 is an infinite set because positive integers greater than 100 are infinite in number.

        (v) The set of prime numbers less than 99 is a finite set because prime numbers less than 99 are finite in number.

        Question 3:

        State whether each of the following set 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 multiple of 5

        (iv) The set of animals living on the earth

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

        Answer:

        (i) The set of lines which are parallel to the x-axis is an infinite set because lines parallel to the x-axis are infinite in number.

        (ii) The set of letters in the English alphabet is a finite set because it has 26 elements.

        (iii) The set of numbers which are multiple of 5 is an infinite set because multiples of 5 are infinite in number.

        (iv) The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (although it is quite a big number).

        (v) The set of circles passing through the origin (0, 0) is an infinite set because infinite number of circles can pass through the origin.

        Page No 9:

        Question 4:

        In the following, state whether A = B or not:

        (i) A = {a, b, c, d}; B = {d, c, b, a}

        (ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}

        (iii) A = {2, 4, 6, 8, 10}; B = {x: x is positive even integer and x ≤ 10}

        (iv) A = {x: x is a multiple of 10}; B = {10, 15, 20, 25, 30 …}

        Answer:

        (i) A = {a, b, c, d}; B = {d, c, b, a}

        The order in which the elements of a set are listed is not significant.

        ∴A = B

        (ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}

        It can be seen that 12 ∈ A but 12 ∉ B.

        ∴A ≠ B

        (iii) A = {2, 4, 6, 8, 10}

        B = {x: x is a positive even integer and x ≤ 10}

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

        ∴A = B

        (iv) A = {x: x is a multiple of 10}

        B = {10, 15, 20, 25, 30 …}

        It can be seen that 15 ∈ B but 15 ∉ A.

        ∴A ≠ B

        Question 5:

        Are the following pair of sets equal? Give reasons.

        (i) A = {2, 3}; B = {x: x is solution of x2 + 5x + 6 = 0}

        (ii) A = {x: x is a letter in the word FOLLOW}; B = {y: y is a letter in the word WOLF}

        Answer:

        (i) A = {2, 3}; B = {x: x is a solution of x2 + 5x + 6 = 0}

        The equation x2 + 5x + 6 = 0 can be solved as: x(x + 3) + 2(x + 3) = 0 (x + 2)(x + 3) = 0 x = –2 or x = –3 ∴A = {2, 3}; B = {–2, –3} ∴A ≠ B

        (ii) A = {x: x is a letter in the word FOLLOW} = {F, O, L, W}

        B = {y: y is a letter in the word WOLF} = {W, O, L, F}

        The order in which the elements of a set are listed is not significant.

        ∴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:

        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}; A = {0, 1}

        It can be seen that

        8 ∈ A, 8 ∉ B, 8 ∉ D, 8 ∉ E, 8 ∉ F, 8 ∉ G, 8 ∉ H

        ⇒ A ≠ B, A ≠ D, A ≠ E, A ≠ F, A ≠ G, A ≠ H

        Also, 2 ∈ A, 2 ∉ C

        ∴ A ≠ C

        3 ∈ B, 3 ∉ C, 3 ∉ E, 3 ∉ F, 3 ∉ G, 3 ∉ H

        ∴ B ≠ C, B ≠ E, B ≠ F, B ≠ G, B ≠ H

        12 ∈ C, 12 ∉ D, 12 ∉ E, 12 ∉ F, 12 ∉ G, 12 ∉ H

        ∴ C ≠ D, C ≠ E, C ≠ F, C ≠ G, C ≠ H

        4 ∈ D, 4 ∉ E, 4 ∉ F, 4 ∉ G, 4 ∉ H

        ∴ D ≠ E, D ≠ F, D ≠ G, D ≠ H

        Similarly, E ≠ F, E ≠ G, E ≠ H

        F ≠ G, F ≠ H, G ≠ H

        The order in which the elements of a set are listed is not significant.

        ∴ B = D and E = G

        Hence, among the given sets, B = D and E = G.

        Page No 12:

        Question 1:

        Make correct statements by filling in the symbols ⊂ or ⊄ 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:

        (i)

        (ii)

        (iii) {x: x is a student of class XI of your school}⊂ {x: x is 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 in a triangle in the same plane}

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

        Page No 13:

        Question 2:

        Examine whether the following statements are true or false:

        (i) {a, b} ⊄ {b, c, a}

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

        (iii) {1, 2, 3} ⊂{1, 3, 5}

        (iv) {a} ⊂ {a. b, c}

        (v) {a} ∈ (a, b, c)

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

        Answer:

        (i) False. Each element of {a, b} is also an element of {b, c, a}.

        (ii) True. a, e are two vowels of the English alphabet.

        (iii) False. 2∈{1, 2, 3}; however, 2∉{1, 3, 5}

        (iv) True. Each element of {a} is also an element of {a, b, c}.

        (v) False. The elements of {a, b, c} are a, b, c. Therefore, {a}⊂{a, b, c}

        (vi) True. {x:x is an even natural number less than 6} = {2, 4}

        {x:x is a natural number which divides 36}= {1, 2, 3, 4, 6, 9, 12, 18, 36}

        Question 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) Φ ∈ A

        (x) Φ ⊂ A

        (xi) {Φ} ⊂ A

        Answer:

        A = {1, 2, {3, 4}, 5}

        (i) The statement {3, 4} ⊂ A is incorrect because 3 ∈ {3, 4}; however, 3∉A.

        (ii) The statement {3, 4} ∈A is correct because {3, 4} is an element of A.

        (iii) The statement {{3, 4}} ⊂ A is correct because {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.

        (iv) The statement 1∈A is correct because 1 is an element of A.

        (v) The statement 1⊂ A is incorrect because an element of a set can never be a subset of itself.

        (vi) The statement {1, 2, 5} ⊂ A is correct because each element of {1, 2, 5} is also an element of A.

        (vii) The statement {1, 2, 5} ∈ A is incorrect because {1, 2, 5} is not an element of A.

        (viii) The statement {1, 2, 3} ⊂ A is incorrect because 3 ∈ {1, 2, 3}; however, 3 ∉ A.

        (ix) The statement Φ ∈ A is incorrect because Φ is not an element of A.

        (x) The statement Φ ⊂ A is correct because Φ is a subset of every set.

        (xi) The statement {Φ} ⊂ A is incorrect because Φ∈ {Φ}; however, Φ ∈ A.

        Question 4:

        Write down all the subsets of the following sets:

        (i) {a}

        (ii) {a, b}

        (iii) {1, 2, 3}

        (iv) Φ

        Answer:

        (i) The subsets of {a} are Φ and {a}.

        (ii) The subsets of {a, b} areΦ, {a}, {b}, and {a, b}.

        (iii) The subsets of {1, 2, 3} areΦ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and

        {1, 2, 3}

        (iv) The only subset of Φ isΦ.

        Question 5:

        How many elements has P(A), if A = Φ?

        Answer:

        We know that if A is a set with m elements i.e., n(A) = m, then n[P(A)] = 2m.

        If A = Φ, then n(A) = 0.

        ∴ n[P(A)] = 20 = 1

        Hence, P(A) has one element.

        Question 6:

        Write the following as intervals:

        (i) {x: x ∈ R, –4 < x ≤ 6}

        (ii) {x: x ∈ R, –12 < x < –10}

        (iii) {x: x ∈ R, 0 ≤ x < 7}

        (iv) {x: x ∈ R, 3 ≤ x ≤ 4}

        Answer:

        (i) {x: x ∈ R, –4 < x ≤ 6} = (–4, 6]

        (ii) {x: x ∈ R, –12 < x < –10} = (–12, –10)

        (iii) {x: x ∈ R, 0 ≤ x < 7} = [0, 7)

        (iv) {x: x ∈ R, 3 ≤ x ≤ 4} = [3, 4]

        Question 7:

        Write the following intervals in set-builder form:

        (i) (–3, 0)

        (ii) [6, 12]

        (iii) (6, 12]

        (iv) [–23, 5)

        Answer:

        (i) (–3, 0) = {x: x ∈ R, –3 < x < 0}

        (ii) [6, 12] = {x: x ∈ R, 6 ≤ x ≤ 12}

        (iii) (6, 12] ={x: x ∈ R, 6 < x ≤ 12}

        (iv) [–23, 5) = {x: x ∈ R, –23 ≤ x < 5}

        Question 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

        Answer:

        (i) For the set of right triangles, the universal set can be the set of triangles or the set of polygons.

        (ii) For the set of isosceles triangles, the universal set can be the set of triangles or the set of polygons or the set of two-dimensional figures.

        Question 9:

        Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universals set (s) for all the three sets A, B and C

        (i) {0, 1, 2, 3, 4, 5, 6}

        (ii) Φ

        (iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

        (iv) {1, 2, 3, 4, 5, 6, 7, 8}

        Answer:

        (i) It can be seen that A ⊂ {0, 1, 2, 3, 4, 5, 6}

        B ⊂ {0, 1, 2, 3, 4, 5, 6}

        However, C ⊄ {0, 1, 2, 3, 4, 5, 6}

        Therefore, the set {0, 1, 2, 3, 4, 5, 6} cannot be the universal set for the sets A, B, and C.

        (ii) A ⊄ Φ, B ⊄ Φ, C ⊄ Φ

        Therefore, Φ cannot be the universal set for the sets A, B, and C.

        (iii) A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

        B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

        C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

        Therefore, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for the sets A, B, and C.

        (iv) A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

        B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

        However, C ⊄ {1, 2, 3, 4, 5, 6, 7, 8}

        Therefore, the set {1, 2, 3, 4, 5, 6, 7, 8} cannot be the universal set for the sets A, B, and C.

        Page No 17:

        Question 1:

        Find the union of each of the following pairs of sets:

        (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 ≤ 6}

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

        (v) A = {1, 2, 3}, B = Φ

        Answer:

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

        X∪ Y= {1, 2, 3, 5}

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

        A∪ B = {a, b, c, e, i, o, u}

        (iii) A = {x: x is a natural number and multiple of 3} = {3, 6, 9 …}

        As B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5, 6}

        A ∪ B = {1, 2, 4, 5, 3, 6, 9, 12 …}

        ∴ A ∪ B = {x: x = 1, 2, 4, 5 or a multiple of 3}

        (iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}

        B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}

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

        ∴ A∪ B = {x: x ∈ N and 1 < x < 10}

        (v) A = {1, 2, 3}, B = Φ

        A∪ B = {1, 2, 3}

        Question 2:

        Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?

        Answer:

        Here, A = {a, b} and B = {a, b, c}

        Yes, A ⊂ B.

        A∪ B = {a, b, c} = B

        Question 3:

        If A and B are two sets such that A ⊂ B, then what is A ∪ B?

        Answer:

        If A and B are two sets such that A ⊂ B, then A ∪ B = B.

        Question 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

        Answer:

        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, 5, 6}

        (ii) A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

        (iii) B ∪ C = {3, 4, 5, 6, 7, 8}

        (iv) B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

        (v) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

        (vi) A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

        (vii) B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

        Page No 18:

        Question 5:

        Find the intersection of each pair of sets:

        (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 ≤ 6}

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

        (v) A = {1, 2, 3}, B = Φ

        Answer:

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

        X ∩ Y = {1, 3}

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

        A ∩ B = {a}

        (iii) A = {x: x is a natural number and multiple of 3} = (3, 6, 9 …}

        B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5}

        ∴ A ∩ B = {3}

        (iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}

        B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}

        A ∩ B = Φ

        (v) A = {1, 2, 3}, B = Φ

        A ∩ B = Φ

        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 ∩ 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)

        Answer:

        (i) A ∩ B = {7, 9, 11}

        (ii) B ∩ C = {11, 13}

        (iii) A ∩ C ∩ D = { A ∩ C} ∩ D = {11} ∩ {15, 17} = Φ

        (iv) A ∩ C = {11}

        (v) B ∩ D = Φ

        (vi) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

        = {7, 9, 11} ∪ {11} = {7, 9, 11}

        (vii) A ∩ D = Φ

        (viii) A ∩ (B ∪ D) = (A ∩ B) ∪ (A ∩ D)

        = {7, 9, 11} ∪ Φ = {7, 9, 11}

        (ix) (A ∩ B) ∩ (B ∪ C) = {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, 15}

        = {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 ∩ B

        (ii) A ∩ C

        (iii) A ∩ D

        (iv) B ∩ C

        (v) B ∩ D

        (vi) C ∩ D

        Answer:

        A = {x: x is a natural number} = {1, 2, 3, 4, 5 …}

        B ={x: x is an even natural number} = {2, 4, 6, 8 …}

        C = {x: x is an odd natural number} = {1, 3, 5, 7, 9 …}

        D = {x: x is a prime number} = {2, 3, 5, 7 …}

        (i) A ∩B = {x: x is a even natural number} = B

        (ii) A ∩ C = {x: x is an odd natural number} = C

        (iii) A ∩ D = {x: x is a prime number} = D

        (iv) B ∩ C = Φ

        (v) B ∩ D = {2}

        (vi) C ∩ D = {x: x is odd prime number}

        Question 8:

        Which of the following pairs of sets are disjoint

        (i) {1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x ≤ 6}

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

        (iii) {x: x is an even integer} and {x: x is an odd integer}

        Answer:

        (i) {1, 2, 3, 4}

        {x: x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}

        Now, {1, 2, 3, 4} ∩ {4, 5, 6} = {4}

        Therefore, this pair of sets is not disjoint.

        (ii) {a, e, i, o, u} ∩ (c, d, e, f} = {e}

        Therefore, {a, e, i, o, u} and (c, d, e, f} are not disjoint.

        (iii) {x: x is an even integer} ∩ {x: x is an odd integer} = Φ

        Therefore, this pair of sets is disjoint.

        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:

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

        (ii) A – C = {3, 9, 15, 18, 21}

        (iii) A – D = {3, 6, 9, 12, 18, 21}

        (iv) B – A = {4, 8, 16, 20}

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

        (vi) D – A = {5, 10, 20}

        (vii) B – C = {20}

        (viii) B – D = {4, 8, 12, 16}

        (ix) C – B = {2, 6, 10, 14}

        (x) D – B = {5, 10, 15}

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

        (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 ∩ Y

        Answer:

        (i) X – Y = {a, c}

        (ii) Y – X = {f, g}

        (iii) X ∩ 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

        Therefore, R – Q is a set of irrational numbers.

        Question 12:

        State whether each of the following statement is true or false. Justify your answer.

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

        (ii) {a, e, i, o, u } and {a, b, c, d} 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.

        Answer:

        (i) False

        As 3 ∈ {2, 3, 4, 5}, 3 ∈ {3, 6}

        ⇒ {2, 3, 4, 5} ∩ {3, 6} = {3}

        (ii) False

        As a ∈ {a, e, i, o, u}, a ∈ {a, b, c, d}

        ⇒ {a, e, i, o, u } ∩ {a, b, c, d} = {a}

        (iii) True

        As {2, 6, 10, 14} ∩ {3, 7, 11, 15} = Φ

        (iv) True

        As {2, 6, 10} ∩ {3, 7, 11} = Φ

        Page No 20:

        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) 

        (ii) 

        (iii)

        (iv)

        (v) 

        (vi) 

        Answer:

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

        A = {1, 2, 3, 4}

        B = {2, 4, 6, 8}

        C = {3, 4, 5, 6}

        (i)

        (ii)

        (iii)

        (iv)

        (v)

        (vi)

        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}

        (ii) B = {d, e, f, g}

        (iii) C = {a, c, e, g}

        (iv) D = {f, g, h, a}

        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}

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

        (vii) {x: x is perfect cube}

        (viii) {x: x + 5 = 8}

        (ix) {x: 2x + 5 = 9}

        (x) {x: x ≥ 7}

        (xi) {x: x ∈ N and 2x + 1 > 10}

        Answer:

        U = N: Set of natural numbers

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

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

        (iii) {x: x is a positive multiple of 3}´ = {x: x ∈ N and x is not a multiple of 3}

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

        (v) {x: x is a natural number divisible by 3 and 5}´ = {x: x is a natural number that is not divisible by 3 or 5}

        (vi) {x: x is a perfect square}´ = {x: x ∈ N and x is not a perfect square}

        (vii) {x: x is a perfect cube}´ = {x: x ∈ N and x is not a perfect cube}

        (viii) {x: x + 5 = 8}´ = {x: x ∈ N and x ≠ 3}

        (ix) {x: 2x + 5 = 9}´ = {x: x ∈ N and x ≠ 2}

        (x) {x: x ≥ 7}´ = {x: x ∈ N and x < 7}

        (xi) {x: x ∈ N and 2x + 1 > 10}´ = {x: x ∈ N and x ≤ 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)  (ii)

        Answer:

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

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

        (i)

        (ii)

        Question 5:

        Draw appropriate Venn diagram for each of the following:

        (i)

        (ii)

        (iii)

        (iv)

        Answer:

        (i)

        (ii)

        (iii)

        (iv)

        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°, what is?

        Answer:

        is the set of all equilateral triangles.

        Page No 21:

        Question 7:

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

        (i) 

        (ii) Φ′ ∩ A = …

        (iii)

        (iv)

        Answer:

        (i)

        (ii) Φ′ ∩ A = U ∩ A = A

        ∴ Φ′ ∩ A = A

        (iii) A ∩ A′ = Φ

        (iv) U′ ∩ A = Φ ∩ A = Φ

        ∴ U′ ∩ A = Φ

        Page No 24:

        Question 1:

        If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩Y).

        Answer:

        It is given that:

        n(X) = 17, n(Y) = 23, n(X ∪ Y) = 38

        n(X ∩ Y) = ?

        We know that:

        Question 2:

        If X and Y are two sets such that X ∪Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X ∩Y have?

        Answer:

        It is given that:

        n(X ∩ Y) = ?

        We know that:

        Question 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?

        Answer:

        Let H be the set of people who speak Hindi, and

        E be the set of people who speak English

        ∴ n(H ∪ E) = 400, n(H) = 250, n(E) = 200

        n(H ∩ E) = ?

        We know that:

        n(H ∪ E) = n(H) + n(E) – n(H ∩ E)

        ∴ 400 = 250 + 200 – n(H ∩ E)

        ⇒ 400 = 450 – n(H ∩ E)

        ⇒ n(H ∩ E) = 450 – 400

        ∴ n(H ∩ E) = 50

        Thus, 50 people can speak both Hindi and English.

        Question 4:

        If S and T are two sets such that S has 21 elements, T has 32 elements, and

        S ∩ T has 11 elements, how many elements does S ∪ T have?

        Answer:

        It is given that:

        n(S) = 21, n(T) = 32, n(S ∩ T) = 11

        We know that:

        n (S ∪ T) = n (S) + n (T) – n (S ∩ T)

        ∴ n (S ∪ T) = 21 + 32 – 11 = 42

        Thus, the set (S ∪ T) has 42 elements.

        Question 5:

        If X and Y are two sets such that X has 40 elements, X ∪Y has 60 elements and X ∩Y has 10 elements, how many elements does Y have?

        Answer:

        It is given that:

        n(X) = 40, n(X ∪ Y) = 60, n(X ∩ Y) = 10

        We know that:

        n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)

        ∴ 60 = 40 + n(Y) – 10

        ∴ n(Y) = 60 – (40 – 10) = 30

        Thus, the set Y has 30 elements.

        Question 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?

        Answer:

        Let C denote the set of people who like coffee, and

        T denote the set of people who like tea

        n(C ∪ T) = 70, n(C) = 37, n(T) = 52

        We know that:

        n(C ∪ T) = n(C) + n(T) – n(C ∩ T)

        ∴ 70 = 37 + 52 – n(C ∩ T)

        ⇒ 70 = 89 – n(C ∩ T)

        ⇒ n(C ∩ T) = 89 – 70 = 19

        Thus, 19 people like both coffee and tea.

        Question 7:

        In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

        Answer:

        Let C denote the set of people who like cricket, and

        T denote the set of people who like tennis

        ∴ n(C ∪ T) = 65, n(C) = 40, n(C ∩ T) = 10

        We know that:

        n(C ∪ T) = n(C) + n(T) – n(C ∩ T)

        ∴ 65 = 40 + n(T) – 10

        ⇒ 65 = 30 + n(T)

        ⇒ n(T) = 65 – 30 = 35

        Therefore, 35 people like tennis.

        Now,

        (T – C) ∪ (T ∩ C) = T

        Also,

        (T – C) ∩ (T ∩ C) = Φ

        ∴ n (T) = n (T – C) + n (T ∩ C)

        ⇒ 35 = n (T – C) + 10

        ⇒ n (T – C) = 35 – 10 = 25

        Thus, 25 people like only tennis.

        Question 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?

        Answer:

        Let F be the set of people in the committee who speak French, and

        S be the set of people in the committee who speak Spanish

        ∴ n(F) = 50, n(S) = 20, n(S ∩ F) = 10

        We know that:

        n(S ∪ F) = n(S) + n(F) – n(S ∩ F)

        = 20 + 50 – 10

        = 70 – 10 = 60

        Thus, 60 people in the committee speak at least one of the two languages.

        Page No 26:

        Question 1:

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

        A = {x: x ∈ R and x satisfy x2 – 8x + 12 = 0},

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

        Answer:

        A = {x: x ∈ R and x satisfies x2 – 8x + 12 = 0}

        2 and 6 are the only solutions of x2 – 8x + 12 = 0.

        ∴ A = {2, 6}

        B = {2, 4, 6}, C = {2, 4, 6, 8 …}, D = {6}

        ∴ D ⊂ A ⊂ B ⊂ C

        Hence, A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B, D ⊂ C

        Question 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 x ∈ A and A ∈ B, then x ∈ 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 x ∈ A and A ⊄ B, then x ∈ B

        (vi) If A ⊂ B and x ∉ B, then x ∉ A

        Answer:

        (i) False

        Let A = {1, 2} and B = {1, {1, 2}, {3}}

        Now,

        ∴ A ∈ B

        However,

        (ii) False

        Let

        As A ⊂ B

        B ∈ C

        However,

        (iii) True

        Let A ⊂ B and B ⊂ C.

        Let x ∈ A

        ∴ A ⊂ C

        (iv) False

        Let

        Accordingly,and .

        However, A ⊂ C

        (v) False

        Let A = {3, 5, 7} and B = {3, 4, 6}

        Now, 5 ∈ A and A ⊄ B

        However, 5 ∉ B

        (vi) True

        Let A ⊂ B and x ∉ B.

        To show: x ∉ A

        If possible, suppose x ∈ A.

        Then, x ∈ B, which is a contradiction as x ∉ B

        ∴x ∉ A

        Question 3:

        Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C.

        Answer:

        Let, A, B and C be the sets such that  and.

        To show: B = C

        Let x ∈ B

        Case I

        x ∈ A

        Also, x ∈ B

        ∴

        ∴ x ∈ A and x ∈ C

        ∴ x ∈ C

        ∴ B ⊂ C

        Similarly, we can show that C ⊂ B.

        ∴ B = C

        Question 4:

        Show that the following four conditions are equivalent:

        (i) A ⊂ B (ii) A – B = Φ

        (iii) A ∪ B = B (iv) A ∩ B = A

        Answer:

        First, we have to show that (i) ⇔ (ii).

        Let A ⊂ B

        To show: A – B ≠ Φ

        If possible, suppose A – B ≠ Φ

        This means that there exists x ∈ A, x ≠ B, which is not possible as A ⊂ B.

        ∴ A – B = Φ

        ∴ A ⊂ B ⇒ A – B = Φ

        Let A – B = Φ

        To show: A ⊂ B

        Let x ∈ A

        Clearly, x ∈ B because if x ∉ B, then A – B ≠ Φ

        ∴ A – B = Φ ⇒ A ⊂ B

        ∴ (i) ⇔ (ii)

        Let A ⊂ B

        To show:

        Clearly,

        Let

        Case I: x ∈ A

        ∴

        Case II: x ∈ B

        Then,

        Conversely, let

        Let x ∈ A

        ∴ A ⊂ B

        Hence, (i) ⇔ (iii)

        Now, we have to show that (i) ⇔ (iv).

        Let A ⊂ B

        Clearly

        Let x ∈ A

        We have to show that

        As A ⊂ B, x ∈ B

        ∴

        ∴

        Hence, A = A ∩ B

        Conversely, suppose A ∩ B = A

        Let x ∈ A

        ⇒

        ⇒ x ∈ A and x ∈ B

        ⇒ x ∈ B

        ∴ A ⊂ B

        Hence, (i) ⇔ (iv).

        Question 5:

        Show that if A ⊂ B, then C – B ⊂ C – A.

        Answer:

        Let A ⊂ B

        To show: C – B ⊂ C – A

        Let x ∈ C – B

        ⇒ x ∈ C and x ∉ B

        ⇒ x ∈ C and x ∉ A [A ⊂ B]

        ⇒ x ∈ C – A

        ∴ C – B ⊂ C – A

        Question 6:

        Assume that P (A) = P (B). Show that A = B.

        Answer:

        Let P(A) = P(B)

        To show: A = B

        Let x ∈ A

        A ∈ P(A) = P(B)

        ∴ x ∈ C, for some C ∈ P(B)

        Now, C ⊂ B

        ∴ x ∈ B

        ∴ A ⊂ B

        Similarly, B ⊂ A

        ∴ A = B

        Question 7:

        Is it true that for any sets A and B, P (A) ∪ P (B) = P (A ∪ B)? Justify your answer.

        Answer:

        False

        Let A = {0, 1} and B = {1, 2}

        ∴ A ∪ B = {0, 1, 2}

        P(A) = {Φ, {0}, {1}, {0, 1}}

        P(B) = {Φ, {1}, {2}, {1, 2}}

        P(A ∪ B) = {Φ, {0}, {1}, {2}, {0, 1}, {1, 2}, {0, 2}, {0, 1, 2}}

        P(A) ∪ P(B) = {Φ, {0}, {1}, {0, 1}, {2}, {1, 2}}

        ∴ P(A) ∪ P(B) ≠ P(A ∪ B)

        Page No 27:

        Question 8:

        Show that for any sets A and B,

        A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)

        Answer:

        To show: A = (A ∩ B) ∪ (A – B)

        Let x ∈ A

        We have to show that x ∈ (A ∩ B) ∪ (A – B)

        Case I

        x ∈ A ∩ B

        Then, x ∈ (A ∩ B) ⊂ (A ∪ B) ∪ (A – B)

        Case II

        x ∉ A ∩ B

        ⇒ x ∉ A or x ∉ B

        ∴ x ∉ B [x ∉ A]

        ∴ x ∉ A – B ⊂ (A ∪ B) ∪ (A – B)

        ∴ A ⊂ (A ∩ B) ∪ (A – B) … (1)

        It is clear that

        A ∩ B ⊂ A and (A – B) ⊂ A

        ∴ (A ∩ B) ∪ (A – B) ⊂ A … (2)

        From (1) and (2), we obtain

        A = (A ∩ B) ∪ (A – B)

        To prove: A ∪ (B – A) ⊂ A ∪ B

        Let x ∈ A ∪ (B – A)

        ⇒ x ∈ A or x ∈ (B – A)

        ⇒ x ∈ A or (x ∈ B and x ∉ A)

        ⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∉ A)

        ⇒ x ∈ (A ∪ B)

        ∴ A ∪ (B – A) ⊂ (A ∪ B) … (3)

        Next, we show that (A ∪ B) ⊂ A ∪ (B – A).

        Let y ∈ A ∪ B

        ⇒ y ∈ A or y ∈ B

        ⇒ (y ∈ A or y ∈ B) and (y ∈ A or y ∉ A)

        ⇒ y ∈ A or (y ∈ B and y ∉ A)

        ⇒ y ∈ A ∪ (B – A)

        ∴ A ∪ B ⊂ A ∪ (B – A) … (4)

        Hence, from (3) and (4), we obtain A ∪ (B – A) = A ∪B.

        Question 9:

        Using properties of sets show that

        (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A.

        Answer:

        (i) To show: A ∪ (A ∩ B) = A

        We know that

        A ⊂ A

        A ∩ B ⊂ A

        ∴ A ∪ (A ∩ B) ⊂ A … (1)

        Also,  A ⊂ A ∪ (A ∩ B) … (2)

        ∴ From (1) and (2), A ∪ (A ∩ B) = A

        (ii) To show: A ∩ (A ∪ B) = A

        A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B)

        = A ∪ (A ∩ B)

        = A {from (1)}

        Question 10:

        Show that A ∩ B = A ∩ C need not imply B = C.

        Answer:

        Let A = {0, 1}, B = {0, 2, 3}, and C = {0, 4, 5}

        Accordingly, A ∩ B = {0} and A ∩ C = {0}

        Here, A ∩ B = A ∩ C = {0}

        However, B ≠ C [2 ∈ B and 2 ∉ C]

        Question 11:

        Let A and B be sets. If A ∩ X = B ∩ X = Φ and A ∪ X = B ∪ X for some set X, show that A = B.

        (Hints A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use distributive law)

        Answer:

        Let A and B be two sets such that A ∩ X = B ∩ X = f and A ∪ X = B ∪ X for some set X.

        To show: A = B

        It can be seen that

        A = A ∩ (A ∪ X) = A ∩ (B ∪ X) [A ∪ X = B ∪ X]

        = (A ∩ B) ∪ (A ∩ X) [Distributive law]

        = (A ∩ B) ∪ Φ [A ∩ X = Φ]

        = A ∩ B … (1)

        Now, B = B ∩ (B ∪ X)

        = B ∩ (A ∪ X) [A ∪ X = B ∪ X]

        = (B ∩ A) ∪ (B ∩ X) [Distributive law]

        = (B ∩ A) ∪ Φ [B ∩ X = Φ]

        = B ∩ A

        = A ∩ B … (2)

        Hence, from (1) and (2), we obtain A = B.

        Question 12:

        Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = Φ.

        Answer:

        Let A = {0, 1}, B = {1, 2}, and C = {2, 0}.

        Accordingly, A ∩ B = {1}, B ∩ C = {2}, and A ∩ C = {0}.

        ∴ A ∩ B, B ∩ C, and A ∩ C are non-empty.

        However, A ∩ B ∩ C = Φ

        Question 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?

        Answer:

        Let U be the set of all students who took part in the survey.

        Let T be the set of students taking tea.

        Let C be the set of students taking coffee.

        Accordingly, n(U) = 600, n(T) = 150, n(C) = 225, n(T ∩ C) = 100

        To find: Number of student taking neither tea nor coffee i.e., we have to find n(T’ ∩ C’).

        n(T’ ∩ C’) = n(T ∪ C)’

        = n(U) – n(T ∪ C)

        = n(U) – [n(T) + n(C) – n(T ∩ C)]

        = 600 – [150 + 225 – 100]

        = 600 – 275

        = 325

        Hence, 325 students were taking neither tea nor coffee.

        Question 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?

        Answer:

        Let U be the set of all students in the group.

        Let E be the set of all students who know English.

        Let H be the set of all students who know Hindi.

        ∴ H ∪ E = U

        Accordingly, n(H) = 100 and n(E) = 50

        = 25

        n(U) = n(H) + – n(H ∩ E)

        = 100 + 50 – 25

        = 125

        Hence, there are 125 students in the group.

        Question 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 read all three newspapers. Find:

        (i) the number of people who read at least one of the newspapers.

        (ii) the number of people who read exactly one newspaper.

        Answer:

        Let A be the set of people who read newspaper H.

        Let B be the set of people who read newspaper T.

        Let C be the set of people who read newspaper I.

        Accordingly, n(A) = 25, n(B) = 26, and n(C) = 26

        n(A ∩ C) = 9, n(A ∩ B) = 11, and n(B ∩ C) = 8

        n(A ∩ B ∩ C) = 3

        Let U be the set of people who took part in the survey.

        (i) Accordingly,

        n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)

        = 25 + 26 + 26 – 11 – 8 – 9 + 3

        = 52

        Hence, 52 people read at least one of the newspapers.

        (ii) Let a be the number of people who read newspapers H and T only.

        Let b denote the number of people who read newspapers I and H only.

        Let c denote the number of people who read newspapers T and I only.

        Let d denote the number of people who read all three newspapers.

        Accordingly, d = n(A ∩ B ∩ C) = 3

        Now, n(A ∩ B) = a + d

        n(B ∩ C) = c + d

        n(C ∩ A) = b + d

        ∴ a + d + c + d + b + d = 11 + 8 + 9 = 28

        ⇒ a + b + c + d = 28 – 2d = 28 – 6 = 22

        Hence, (52 – 22) = 30 people read exactly one newspaper.

        Question 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.

        Answer:

        Let A, B, and C be the set of people who like product A, product B, and product C respectively.

        Accordingly, n(A) = 21, n(B) = 26, n(C) = 29, n(A ∩ B) = 14, n(C ∩ A) = 12,

        n(B ∩ C) = 14, n(A ∩ B ∩ C) = 8

        The Venn diagram for the given problem can be drawn as

        It can be seen that number of people who like product C only is

        {29 – (4 + 8 + 6)} = 11

         

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