Mathematics · Textbook solutions

Sets

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 226 questions

1.2 Sets and their Representations

38 q

Solved Examples

Worked · 5
  1. 1.1 Eg.1
    Write the solution set of the equation x2+x−2=0x^2 + x - 2 = 0 in roster form.
  2. 1.1 Eg.2
    Write the set {x:x is a positive integer and x2<40}\{x : x \text{ is a positive integer and } x^2 < 40\} in the roster form.
  3. 1.1 Eg.3
    Write the set A={1,4,9,16,25,…}A = \{1, 4, 9, 16, 25, \ldots\} in set-builder form.
  4. 1.1 Eg.4
    Write the set {12,23,34,45,56,67}\left\{\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}\right\} in the set-builder form.
  5. 1.1 Eg.5
    Match each of the set on the left described in the roster form with the same set on the right described in the set-builder form : (i) {P,R,I,N,C,A,L}\{P, R, I, N, C, A, L\} (ii) {0}\{0\} (iii) {1,2,3,6,9,18}\{1, 2, 3, 6, 9, 18\} (iv) {3,−3}\{3, -3\} (a) {x:x is a positive integer and is a divisor of 18}\{x : x \text{ is a positive integer and is a divisor of } 18\} (b) {x:x is an integer and x2−9=0}\{x : x \text{ is an integer and } x^2 - 9 = 0\} (c) {x:x is an integer and x+1=1}\{x : x \text{ is an integer and } x + 1 = 1\} (d) {x:x is a letter of the word PRINCIPAL}\{x : x \text{ is a letter of the word PRINCIPAL}\}

Exercise 1.1

Practice · 33
  1. Which of the following are sets ? Justify your answer.
    Ex 1.1 Q1(i)
    The collection of all the months of a year beginning with the letter J.
  2. Ex 1.1 Q1(ii)
    The collection of ten most talented writers of India.
  3. Ex 1.1 Q1(iii)
    A team of eleven best-cricket batsmen of the world.
  4. Ex 1.1 Q1(iv)
    The collection of all boys in your class.
  5. Ex 1.1 Q1(v)
    The collection of all natural numbers less than 100.
  6. Ex 1.1 Q1(vi)
    A collection of novels written by the writer Munshi Prem Chand.
  7. Ex 1.1 Q1(vii)
    The collection of all even integers.
  8. Ex 1.1 Q1(viii)
    The collection of questions in this Chapter.
  9. Ex 1.1 Q1(ix)
    A collection of most dangerous animals of the world.
  10. Let A={1,2,3,4,5,6}A = \{1, 2, 3, 4, 5, 6\}. Insert the appropriate symbol ∈\in or ∉\notin in the blank spaces:
    Ex 1.1 Q2(i)
    5…A5 \ldots A
  11. Ex 1.1 Q2(ii)
    8…A8 \ldots A
  12. Ex 1.1 Q2(iii)
    0…A0 \ldots A
  13. Ex 1.1 Q2(iv)
    4…A4 \ldots A
  14. Ex 1.1 Q2(v)
    2…A2 \ldots A
  15. Ex 1.1 Q2(vi)
    10…A10 \ldots A
  16. Write the following sets in roster form:
    Ex 1.1 Q3(i)
    A={x:x is an integer and −3≤x<7}A = \{x : x \text{ is an integer and } -3 \le x < 7\}
  17. Ex 1.1 Q3(ii)
    B={x:x is a natural number less than 6}B = \{x : x \text{ is a natural number less than } 6\}
  18. Ex 1.1 Q3(iii)
    C={x:x is a two-digit natural number such that the sum of its digits is 8}C = \{x : x \text{ is a two-digit natural number such that the sum of its digits is } 8\}
  19. Ex 1.1 Q3(iv)
    D={x:x is a prime number which is divisor of 60}D = \{x : x \text{ is a prime number which is divisor of } 60\}
  20. Ex 1.1 Q3(v)
    E = The set of all letters in the word TRIGONOMETRY
  21. Ex 1.1 Q3(vi)
    F = The set of all letters in the word BETTER
  22. Write the following sets in the set-builder form :
    Ex 1.1 Q4(i)
    (3,6,9,12}(3, 6, 9, 12\}
  23. Ex 1.1 Q4(ii)
    {2,4,8,16,32}\{2, 4, 8, 16, 32\}
  24. Ex 1.1 Q4(iii)
    {5,25,125,625}\{5, 25, 125, 625\}
  25. Ex 1.1 Q4(iv)
    {2,4,6,…}\{2, 4, 6, \ldots\}
  26. Ex 1.1 Q4(v)
    {1,4,9,…,100}\{1, 4, 9, \ldots, 100\}
  27. List all the elements of the following sets :
    Ex 1.1 Q5(i)
    A={x:x is an odd natural number}A = \{x : x \text{ is an odd natural number}\}
  28. Ex 1.1 Q5(ii)
    B={x:x is an integer,−12<x<92}B = \left\{x : x \text{ is an integer}, -\frac{1}{2} < x < \frac{9}{2}\right\}
  29. Ex 1.1 Q5(iii)
    C={x:x is an integer,x2≤4}C = \{x : x \text{ is an integer}, x^2 \le 4\}
  30. Ex 1.1 Q5(iv)
    D={x:x is a letter in the word "LOYAL"}D = \{x : x \text{ is a letter in the word "LOYAL"}\}
  31. Ex 1.1 Q5(v)
    E={x:x is a month of a year not having 31 days}E = \{x : x \text{ is a month of a year not having } 31 \text{ days}\}
  32. Ex 1.1 Q5(vi)
    F={x:x is a consonant in the English alphabet which precedes k}F = \{x : x \text{ is a consonant in the English alphabet which precedes } k\}.
  33. Ex 1.1 Q6
    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}\{1, 2, 3, 6\} (ii) {2,3}\{2, 3\} (iii) {M,A,T,H,E,I,C,S}\{M, A, T, H, E, I, C, S\} (iv) {1,3,5,7,9}\{1, 3, 5, 7, 9\} (a) {x:x is a prime number and a divisor of 6}\{x : x \text{ is a prime number and a divisor of } 6\} (b) {x:x is an odd natural number less than 10}\{x : x \text{ is an odd natural number less than } 10\} (c) {x:x is natural number and divisor of 6}\{x : x \text{ is natural number and divisor of } 6\} (d) {x:x is a letter of the word MATHEMATICS}\{x : x \text{ is a letter of the word MATHEMATICS}\}.

1.3-1.5 The Empty, Finite and Equal Sets

24 q

Solved Examples

Worked · 3
  1. 1.2 Eg.6
    State which of the following sets are finite or infinite : (i) {x:x∈N and (x−1)(x−2)=0}\{x : x \in \mathbf{N} \text{ and } (x - 1)(x - 2) = 0\} (ii) {x:x∈N and x2=4}\{x : x \in \mathbf{N} \text{ and } x^2 = 4\} (iii) {x:x∈N and 2x−1=0}\{x : x \in \mathbf{N} \text{ and } 2x - 1 = 0\} (iv) {x:x∈N and x is prime}\{x : x \in \mathbf{N} \text{ and } x \text{ is prime}\} (v) {x:x∈N and x is odd}\{x : x \in \mathbf{N} \text{ and } x \text{ is odd}\}
  2. 1.2 Eg.7
    Find the pairs of equal sets, if any, give reasons: A={0}A = \{0\}, B={x:x>15 and x<5}B = \{x : x > 15 \text{ and } x < 5\}, C={x:x−5=0}C = \{x : x - 5 = 0\}, D={x:x2=25}D = \{x : x^2 = 25\}, E={x:x is an integral positive root of the equation x2−2x−15=0}E = \{x : x \text{ is an integral positive root of the equation } x^2 - 2x - 15 = 0\}.
  3. 1.2 Eg.8
    Which of the following pairs of sets are equal? Justify your answer. (i) X, the set of letters in "ALLOY" and B, the set of letters in "LOYAL". (ii) A={n:n∈Z and n2≤4}A = \{n : n \in \mathbf{Z} \text{ and } n^2 \le 4\} and B={x:x∈R and x2−3x+2=0}B = \{x : x \in \mathbf{R} \text{ and } x^2 - 3x + 2 = 0\}.

Exercise 1.2

Practice · 21
  1. Which of the following are examples of the null set
    Ex 1.2 Q1(i)
    Set of odd natural numbers divisible by 2
  2. Ex 1.2 Q1(ii)
    Set of even prime numbers
  3. Ex 1.2 Q1(iii)
    {x:x is a natural numbers,x<5 and x>7}\{x : x \text{ is a natural numbers}, x < 5 \text{ and } x > 7\}
  4. Ex 1.2 Q1(iv)
    {y:y is a point common to any two parallel lines}\{y : y \text{ is a point common to any two parallel lines}\}
  5. Which of the following sets are finite or infinite
    Ex 1.2 Q2(i)
    The set of months of a year
  6. Ex 1.2 Q2(ii)
    {1,2,3,…}\{1, 2, 3, \ldots\}
  7. Ex 1.2 Q2(iii)
    {1,2,3,…99,100}\{1, 2, 3, \ldots 99, 100\}
  8. Ex 1.2 Q2(iv)
    The set of positive integers greater than 100
  9. Ex 1.2 Q2(v)
    The set of prime numbers less than 99
  10. State whether each of the following set is finite or infinite:
    Ex 1.2 Q3(i)
    The set of lines which are parallel to the xx-axis
  11. Ex 1.2 Q3(ii)
    The set of letters in the English alphabet
  12. Ex 1.2 Q3(iii)
    The set of numbers which are multiple of 5
  13. Ex 1.2 Q3(iv)
    The set of animals living on the earth
  14. Ex 1.2 Q3(v)
    The set of circles passing through the origin (0,0)(0,0)
  15. In the following, state whether A=BA = B or not:
    Ex 1.2 Q4(i)
    A={a,b,c,d}A = \{a, b, c, d\} B={d,c,b,a}B = \{d, c, b, a\}
  16. Ex 1.2 Q4(ii)
    A={4,8,12,16}A = \{4, 8, 12, 16\} B={8,4,16,18}B = \{8, 4, 16, 18\}
  17. Ex 1.2 Q4(iii)
    A={2,4,6,8,10}A = \{2, 4, 6, 8, 10\} B={x:x is positive even integer and x≤10}B = \{x : x \text{ is positive even integer and } x \le 10\}
  18. Ex 1.2 Q4(iv)
    A={x:x is a multiple of 10}A = \{x : x \text{ is a multiple of } 10\}, B={10,15,20,25,30,…}B = \{10, 15, 20, 25, 30, \ldots\}
  19. Are the following pair of sets equal ? Give reasons.
    Ex 1.2 Q5(i)
    A={2,3}A = \{2, 3\}, B={x:x is solution of x2+5x+6=0}B = \{x : x \text{ is solution of } x^2 + 5x + 6 = 0\}
  20. Ex 1.2 Q5(ii)
    A={x:x is a letter in the word FOLLOW}A = \{x : x \text{ is a letter in the word FOLLOW}\} B={y:y is a letter in the word WOLF}B = \{y : y \text{ is a letter in the word WOLF}\}
  21. Ex 1.2 Q6
    From the sets given below, select equal sets : A={2,4,8,12}A = \{2, 4, 8, 12\}, B={1,2,3,4}B = \{1, 2, 3, 4\}, C={4,8,12,14}C = \{4, 8, 12, 14\}, D={3,1,4,2}D = \{3, 1, 4, 2\}, E={−1,1}E = \{-1, 1\}, F={0,a}F = \{0, a\}, G={1,−1}G = \{1, -1\}, H={0,1}H = \{0, 1\}

1.6 Subsets

45 q

Solved Examples

Worked · 3
  1. 1.3 Eg.9
    Consider the sets ϕ\phi, A={1,3}A = \{ 1, 3 \}, B={1,5,9}B = \{1, 5, 9\}, C={1,3,5,7,9}C = \{1, 3, 5, 7, 9\}. Insert the symbol ⊂\subset or ⊄\not\subset between each of the following pair of sets: (i) ϕ…B\phi \ldots B (ii) A…BA \ldots B (iii) A…CA \ldots C (iv) B…CB \ldots C
  2. 1.3 Eg.10
    Let A={a,e,i,o,u}A = \{ a, e, i, o, u\} and B={a,b,c,d}B = \{ a, b, c, d\}. Is AA a subset of BB ? No. (Why?). Is BB a subset of AA? No. (Why?)
  3. 1.3 Eg.11
    Let AA, BB and CC be three sets. If A∈BA \in B and B⊂CB \subset C, is it true that A⊂CA \subset C?. If not, give an example.

Exercise 1.3

Practice · 42
  1. Make correct statements by filling in the symbols ⊂\subset or ⊄\not\subset in the blank spaces :
    Ex 1.3 Q1(i)
    {2,3,4}…{1,2,3,4,5}\{ 2, 3, 4 \} \ldots \{ 1, 2, 3, 4, 5 \}
  2. Ex 1.3 Q1(ii)
    {a,b,c}…{b,c,d}\{ a, b, c \} \ldots \{ b, c, d \}
  3. Ex 1.3 Q1(iii)
    {x:x is a student of Class XI of your school}…{x:x student of your school}\{ x : x \text{ is a student of Class XI of your school} \} \ldots \{ x : x \text{ student of your school} \}
  4. Ex 1.3 Q1(iv)
    {x:x is a circle in the plane}…{x:x is a circle in the same plane with radius 1 unit}\{ x : x \text{ is a circle in the plane} \} \ldots \{ x : x \text{ is a circle in the same plane with radius 1 unit} \}
  5. Ex 1.3 Q1(v)
    {x:x is a triangle in a plane}…{x:x is a rectangle in the plane}\{ x : x \text{ is a triangle in a plane} \} \ldots \{ x : x \text{ is a rectangle in the plane} \}
  6. Ex 1.3 Q1(vi)
    {x:x is an equilateral triangle in a plane}…{x:x is a triangle in the same plane}\{ x : x \text{ is an equilateral triangle in a plane} \} \ldots \{ x : x \text{ is a triangle in the same plane} \}
  7. Ex 1.3 Q1(vii)
    {x:x is an even natural number}…{x:x is an integer}\{ x : x \text{ is an even natural number} \} \ldots \{ x : x \text{ is an integer} \}
  8. Examine whether the following statements are true or false:
    Ex 1.3 Q2(i)
    {a,b}⊄{b,c,a}\{ a, b \} \not\subset \{ b, c, a \}
  9. Ex 1.3 Q2(ii)
    {a,e}⊂{x:x is a vowel in the English alphabet}\{ a, e \} \subset \{ x : x \text{ is a vowel in the English alphabet} \}
  10. Ex 1.3 Q2(iii)
    {1,2,3}⊂{1,3,5}\{ 1, 2, 3 \} \subset \{ 1, 3, 5 \}
  11. Ex 1.3 Q2(iv)
    {a}⊂{a,b,c}\{ a \} \subset \{ a, b, c \}
  12. Ex 1.3 Q2(v)
    {a}∈{a,b,c}\{ a \} \in \{ a, b, c \}
  13. Ex 1.3 Q2(vi)
    {x:x is an even natural number less than 6}⊂{x:x is a natural number which divides 36}\{ x : x \text{ is an even natural number less than } 6 \} \subset \{ x : x \text{ is a natural number which divides } 36 \}
  14. Let A={1,2,{3,4},5}A = \{ 1, 2, \{ 3, 4 \}, 5 \}. Which of the following statements are incorrect and why?
    Ex 1.3 Q3(i)
    {3,4}⊂A\{3, 4\} \subset A
  15. Ex 1.3 Q3(ii)
    {3,4}∈A\{3, 4\} \in A
  16. Ex 1.3 Q3(iii)
    {{3,4}}⊂A\{\{3, 4\}\} \subset A
  17. Ex 1.3 Q3(iv)
    1∈A1 \in A
  18. Ex 1.3 Q3(v)
    1⊂A1 \subset A
  19. Ex 1.3 Q3(vi)
    {1,2,5}⊂A\{1, 2, 5\} \subset A
  20. Ex 1.3 Q3(vii)
    {1,2,5}∈A\{1, 2, 5\} \in A
  21. Ex 1.3 Q3(viii)
    {1,2,3}⊂A\{1, 2, 3\} \subset A
  22. Ex 1.3 Q3(ix)
    ϕ∈A\phi \in A
  23. Ex 1.3 Q3(x)
    ϕ⊂A\phi \subset A
  24. Ex 1.3 Q3(xi)
    {ϕ}⊂A\{\phi\} \subset A
  25. Write down all the subsets of the following sets
    Ex 1.3 Q4(i)
    {a}\{a\}
  26. Ex 1.3 Q4(ii)
    {a,b}\{a, b\}
  27. Ex 1.3 Q4(iii)
    {1,2,3}\{1, 2, 3\}
  28. Ex 1.3 Q4(iv)
    ϕ\phi
  29. Write the following as intervals :
    Ex 1.3 Q5(i)
    {x:x∈R,−4<x≤6}\{ x : x \in \mathbf{R}, -4 < x \le 6 \}
  30. Ex 1.3 Q5(ii)
    {x:x∈R,−12<x<−10}\{ x : x \in \mathbf{R}, -12 < x < -10 \}
  31. Ex 1.3 Q5(iii)
    {x:x∈R,0≤x<7}\{ x : x \in \mathbf{R}, 0 \le x < 7 \}
  32. Ex 1.3 Q5(iv)
    {x:x∈R,3≤x≤4}\{ x : x \in \mathbf{R}, 3 \le x \le 4 \}
  33. Write the following intervals in set-builder form :
    Ex 1.3 Q6(i)
    (−3,0)(-3, 0)
  34. Ex 1.3 Q6(ii)
    [6,12][6, 12]
  35. Ex 1.3 Q6(iii)
    (6,12](6, 12]
  36. Ex 1.3 Q6(iv)
    [−23,5)[-23, 5)
  37. What universal set(s) would you propose for each of the following :
    Ex 1.3 Q7(i)
    The set of right triangles.
  38. Ex 1.3 Q7(ii)
    The set of isosceles triangles.
  39. Given the sets A={1,3,5}A = \{1, 3, 5\}, B={2,4,6}B = \{2, 4, 6\} and C={0,2,4,6,8}C = \{0, 2, 4, 6, 8\}, which of the following may be considered as universal set (s) for all the three sets AA, BB and CC
    Ex 1.3 Q8(i)
    {0,1,2,3,4,5,6}\{0, 1, 2, 3, 4, 5, 6\}
  40. Ex 1.3 Q8(ii)
    ϕ\phi
  41. Ex 1.3 Q8(iii)
    {0,1,2,3,4,5,6,7,8,9,10}\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}
  42. Ex 1.3 Q8(iv)
    {1,2,3,4,5,6,7,8}\{1, 2, 3, 4, 5, 6, 7, 8\}

1.9 Operations on Sets

62 q

Solved Examples

Worked · 8
  1. 1.4 Eg.12
    Let A={2,4,6,8}A = \{ 2, 4, 6, 8\} and B={6,8,10,12}B = \{ 6, 8, 10, 12\}. Find A∪BA \cup B.
  2. 1.4 Eg.13
    Let A={a,e,i,o,u}A = \{ a, e, i, o, u \} and B={a,i,u}B = \{ a, i, u \}. Show that A∪B=AA \cup B = A.
  3. 1.4 Eg.14
    Let X={Ram, Geeta, Akbar}X = \{\text{Ram, Geeta, Akbar}\} be the set of students of Class XI, who are in school hockey team. Let Y={Geeta, David, Ashok}Y = \{\text{Geeta, David, Ashok}\} be the set of students from Class XI who are in the school football team. Find X∪YX \cup Y and interpret the set.
  4. 1.4 Eg.15
    Consider the sets AA and BB of Example 12. Find A∩BA \cap B.
  5. 1.4 Eg.16
    Consider the sets XX and YY of Example 14. Find X∩YX \cap Y.
  6. 1.4 Eg.17
    Let A={1,2,3,4,5,6,7,8,9,10}A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} and B={2,3,5,7}B = \{ 2, 3, 5, 7 \}. Find A∩BA \cap B and hence show that A∩B=BA \cap B = B.
  7. 1.4 Eg.18
    Let A={1,2,3,4,5,6}A = \{ 1, 2, 3, 4, 5, 6\}, B={2,4,6,8}B = \{ 2, 4, 6, 8 \}. Find A−BA - B and B−AB - A.
  8. 1.4 Eg.19
    Let V={a,e,i,o,u}V = \{ a, e, i, o, u \} and B={a,i,k,u}B = \{ a, i, k, u\}. Find V−BV - B and B−VB - V.

Exercise 1.4

Practice · 54
  1. Find the union of each of the following pairs of sets :
    Ex 1.4 Q1(i)
    X={1,3,5}X = \{1, 3, 5\}, Y={1,2,3}Y = \{1, 2, 3\}
  2. Ex 1.4 Q1(ii)
    A=[a,e,i,o,u}A = [ a, e, i, o, u\}, B={a,b,c}B = \{a, b, c\}
  3. Ex 1.4 Q1(iii)
    A={x:x is a natural number and multiple of 3}A = \{x : x \text{ is a natural number and multiple of } 3\}, B={x:x is a natural number less than 6}B = \{x : x \text{ is a natural number less than } 6\}
  4. Ex 1.4 Q1(iv)
    A={x:x is a natural number and 1<x≤6}A = \{x : x \text{ is a natural number and } 1 < x \le 6 \}, B={x:x is a natural number and 6<x<10}B = \{x : x \text{ is a natural number and } 6 < x < 10 \}
  5. Ex 1.4 Q1(v)
    A={1,2,3}A = \{1, 2, 3\}, B=ϕB = \phi
  6. Ex 1.4 Q2
    Let A={a,b}A = \{ a, b \}, B={a,b,c}B = \{a, b, c\}. Is A⊂BA \subset B ? What is A∪BA \cup B ?
  7. Ex 1.4 Q3
    If AA and BB are two sets such that A⊂BA \subset B, then what is A∪BA \cup B ?
  8. If A={1,2,3,4}A = \{1, 2, 3, 4\}, B={3,4,5,6}B = \{3, 4, 5, 6\}, C={5,6,7,8}C = \{5, 6, 7, 8\} and D={7,8,9,10}D = \{ 7, 8, 9, 10 \}; find
    Ex 1.4 Q4(i)
    A∪BA \cup B
  9. Ex 1.4 Q4(ii)
    A∪CA \cup C
  10. Ex 1.4 Q4(iii)
    B∪CB \cup C
  11. Ex 1.4 Q4(iv)
    B∪DB \cup D
  12. Ex 1.4 Q4(v)
    A∪B∪CA \cup B \cup C
  13. Ex 1.4 Q4(vi)
    A∪B∪DA \cup B \cup D
  14. Ex 1.4 Q4(vii)
    B∪C∪DB \cup C \cup D
  15. Ex 1.4 Q5
    Find the intersection of each pair of sets of question 1 above.
  16. If A={3,5,7,9,11}A = \{ 3, 5, 7, 9, 11 \}, B={7,9,11,13}B = \{7, 9, 11, 13\}, C={11,13,15}C = \{11, 13, 15\} and D={15,17}D = \{15, 17\}; find
    Ex 1.4 Q6(i)
    A∩BA \cap B
  17. Ex 1.4 Q6(ii)
    B∩CB \cap C
  18. Ex 1.4 Q6(iii)
    A∩C∩DA \cap C \cap D
  19. Ex 1.4 Q6(iv)
    A∩CA \cap C
  20. Ex 1.4 Q6(v)
    B∩DB \cap D
  21. Ex 1.4 Q6(vi)
    A∩(B∪C)A \cap (B \cup C)
  22. Ex 1.4 Q6(vii)
    A∩DA \cap D
  23. Ex 1.4 Q6(viii)
    A∩(B∪D)A \cap (B \cup D)
  24. Ex 1.4 Q6(ix)
    (A∩B)∩(B∪C)(A \cap B) \cap (B \cup C)
  25. Ex 1.4 Q6(x)
    (A∪D)∩(B∪C)(A \cup D) \cap (B \cup C)
  26. If A={x:x is a natural number}A = \{x : x \text{ is a natural number}\}, B={x:x is an even natural number}B = \{x : x \text{ is an even natural number}\}, C={x:x is an odd natural number}C = \{x : x \text{ is an odd natural number}\} and D={x:x is a prime number}D = \{x : x \text{ is a prime number}\}, find
    Ex 1.4 Q7(i)
    A∩BA \cap B
  27. Ex 1.4 Q7(ii)
    A∩CA \cap C
  28. Ex 1.4 Q7(iii)
    A∩DA \cap D
  29. Ex 1.4 Q7(iv)
    B∩CB \cap C
  30. Ex 1.4 Q7(v)
    B∩DB \cap D
  31. Ex 1.4 Q7(vi)
    C∩DC \cap D
  32. Which of the following pairs of sets are disjoint
    Ex 1.4 Q8(i)
    {1,2,3,4}\{1, 2, 3, 4\} and {x:x is a natural number and 4≤x≤6}\{x : x \text{ is a natural number and } 4 \le x \le 6 \}
  33. Ex 1.4 Q8(ii)
    {a,e,i,o,u}\{ a, e, i, o, u \} and {c,d,e,f}\{ c, d, e, f \}
  34. Ex 1.4 Q8(iii)
    {x:x is an even integer}\{x : x \text{ is an even integer} \} and {x:x is an odd integer}\{x : x \text{ is an odd integer}\}
  35. If A={3,6,9,12,15,18,21}A = \{3, 6, 9, 12, 15, 18, 21\}, B={4,8,12,16,20}B = \{ 4, 8, 12, 16, 20 \}, C={2,4,6,8,10,12,14,16}C = \{ 2, 4, 6, 8, 10, 12, 14, 16 \}, D={5,10,15,20}D = \{5, 10, 15, 20 \}; find
    Ex 1.4 Q9(i)
    A−BA - B
  36. Ex 1.4 Q9(ii)
    A−CA - C
  37. Ex 1.4 Q9(iii)
    A−DA - D
  38. Ex 1.4 Q9(iv)
    B−AB - A
  39. Ex 1.4 Q9(v)
    C−AC - A
  40. Ex 1.4 Q9(vi)
    D−AD - A
  41. Ex 1.4 Q9(vii)
    B−CB - C
  42. Ex 1.4 Q9(viii)
    B−DB - D
  43. Ex 1.4 Q9(ix)
    C−BC - B
  44. Ex 1.4 Q9(x)
    D−BD - B
  45. Ex 1.4 Q9(xi)
    C−DC - D
  46. Ex 1.4 Q9(xii)
    D−CD - C
  47. If X={a,b,c,d}X = \{ a, b, c, d \} and Y={f,b,d,g}Y = \{ f, b, d, g\}, find
    Ex 1.4 Q10(i)
    X−YX - Y
  48. Ex 1.4 Q10(ii)
    Y−XY - X
  49. Ex 1.4 Q10(iii)
    X∩YX \cap Y
  50. Ex 1.4 Q11
    If R\mathbf{R} is the set of real numbers and Q\mathbf{Q} is the set of rational numbers, then what is R−Q\mathbf{R} - \mathbf{Q}?
  51. State whether each of the following statement is true or false. Justify your answer.
    Ex 1.4 Q12(i)
    {2,3,4,5}\{ 2, 3, 4, 5 \} and {3,6}\{ 3, 6\} are disjoint sets.
  52. Ex 1.4 Q12(ii)
    {a,e,i,o,u}\{ a, e, i, o, u \} and {a,b,c,d}\{ a, b, c, d \} are disjoint sets.
  53. Ex 1.4 Q12(iii)
    {2,6,10,14}\{ 2, 6, 10, 14 \} and {3,7,11,15}\{ 3, 7, 11, 15\} are disjoint sets.
  54. Ex 1.4 Q12(iv)
    {2,6,10}\{ 2, 6, 10 \} and {3,7,11}\{ 3, 7, 11\} are disjoint sets.

1.10 Complement of a Set

35 q

Solved Examples

Worked · 3
  1. 1.5 Eg.20
    Let U={1,2,3,4,5,6,7,8,9,10}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} and A={1,3,5,7,9}A = \{1, 3, 5, 7, 9\}. Find A′A'.
  2. 1.5 Eg.21
    Let UU be universal set of all the students of Class XI of a coeducational school and AA be the set of all girls in Class XI. Find A′A'.
  3. 1.5 Eg.22
    Let U={1,2,3,4,5,6}U = \{1, 2, 3, 4, 5, 6\}, A={2,3}A = \{2, 3\} and B={3,4,5}B = \{3, 4, 5\}. Find A′A', B′B', A′∩B′A' \cap B', A∪BA \cup B and hence show that A∪B‾=A′∩B′\overline{A \cup B} = A' \cap B'.

Exercise 1.5

Practice · 32
  1. Let U={1,2,3,4,5,6,7,8,9}U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}, A={1,2,3,4}A = \{ 1, 2, 3, 4\}, B={2,4,6,8}B = \{ 2, 4, 6, 8 \} and C={3,4,5,6}C = \{ 3, 4, 5, 6 \}. Find
    Ex 1.5 Q1(i)
    A′A'
  2. Ex 1.5 Q1(ii)
    B′B'
  3. Ex 1.5 Q1(iii)
    A∪C‾\overline{A \cup C}
  4. Ex 1.5 Q1(iv)
    A∪B‾\overline{A \cup B}
  5. Ex 1.5 Q1(v)
    (A′)′(A')'
  6. Ex 1.5 Q1(vi)
    (B−C)′(B - C)'
  7. If U={a,b,c,d,e,f,g,h}U = \{ a, b, c, d, e, f, g, h\}, find the complements of the following sets :
    Ex 1.5 Q2(i)
    A={a,b,c}A = \{a, b, c\}
  8. Ex 1.5 Q2(ii)
    B={d,e,f,g}B = \{d, e, f, g\}
  9. Ex 1.5 Q2(iii)
    C={a,c,e,g}C = \{a, c, e, g\}
  10. Ex 1.5 Q2(iv)
    D={f,g,h,a}D = \{ f, g, h, a\}
  11. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
    Ex 1.5 Q3(i)
    {x:x is an even natural number}\{x : x \text{ is an even natural number}\}
  12. Ex 1.5 Q3(ii)
    {x:x is an odd natural number}\{ x : x \text{ is an odd natural number} \}
  13. Ex 1.5 Q3(iii)
    {x:x is a positive multiple of 3}\{x : x \text{ is a positive multiple of } 3\}
  14. Ex 1.5 Q3(iv)
    {x:x is a prime number}\{ x : x \text{ is a prime number} \}
  15. Ex 1.5 Q3(v)
    {x:x is a natural number divisible by 3 and 5}\{x : x \text{ is a natural number divisible by } 3 \text{ and } 5\}
  16. Ex 1.5 Q3(vi)
    {x:x is a perfect square}\{ x : x \text{ is a perfect square} \}
  17. Ex 1.5 Q3(vii)
    {x:x is a perfect cube}\{ x : x \text{ is a perfect cube}\}
  18. Ex 1.5 Q3(viii)
    {x:x+5=8}\{ x : x + 5 = 8 \}
  19. Ex 1.5 Q3(ix)
    {x:2x+5=9}\{ x : 2x + 5 = 9\}
  20. Ex 1.5 Q3(x)
    {x:x≥7}\{ x : x \ge 7 \}
  21. Ex 1.5 Q3(xi)
    {x:x∈N and 2x+1>10}\{ x : x \in \mathbf{N} \text{ and } 2x + 1 > 10 \}
  22. If U={1,2,3,4,5,6,7,8,9}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9 \}, A={2,4,6,8}A = \{2, 4, 6, 8\} and B={2,3,5,7}B = \{ 2, 3, 5, 7\}. Verify that
    Ex 1.5 Q4(i)
    A∪B‾=A′∩B′\overline{A \cup B} = A' \cap B'
  23. Ex 1.5 Q4(ii)
    A∩B‾=A′∪B′\overline{A \cap B} = A' \cup B'
  24. Draw appropriate Venn diagram for each of the following :
    Ex 1.5 Q5(i)
    A∪B‾\overline{A \cup B}
  25. Ex 1.5 Q5(ii)
    A′∩B′A' \cap B'
  26. Ex 1.5 Q5(iii)
    A∩B‾\overline{A \cap B}
  27. Ex 1.5 Q5(iv)
    A′∪B′A' \cup B'
  28. Ex 1.5 Q6
    Let UU be the set of all triangles in a plane. If AA is the set of all triangles with at least one angle different from 60∘60^\circ, what is A′A'?
  29. Fill in the blanks to make each of the following a true statement :
    Ex 1.5 Q7(i)
    A∪A′=…A \cup A' = \ldots
  30. Ex 1.5 Q7(ii)
    ϕ′∩A=…\phi' \cap A = \ldots
  31. Ex 1.5 Q7(iii)
    A∩A′=…A \cap A' = \ldots
  32. Ex 1.5 Q7(iv)
    U′∩A=…U' \cap A = \ldots

Miscellaneous Exercise on Chapter 1

22 q

Miscellaneous Examples

Worked · 3
  1. Misc Eg.23
    Show that the set of letters needed to spell " CATARACT " and the set of letters needed to spell " TRACT" are equal.
  2. Misc Eg.24
    List all the subsets of the set {−1,0,1}\{ -1, 0, 1 \}.
  3. Misc Eg.25
    Show that A∪B=A∩BA \cup B = A \cap B implies A=BA = B.

Miscellaneous Exercise

Practice · 19
  1. Misc Q1
    Decide, among the following sets, which sets are subsets of one and another: A={x:x∈R and x satisfy x2−8x+12=0}A = \{ x : x \in \mathbf{R} \text{ and } x \text{ satisfy } x^{2} - 8x + 12 = 0 \}, B={2,4,6}B = \{ 2, 4, 6 \}, C={2,4,6,8,…}C = \{ 2, 4, 6, 8, \ldots \}, D={6}D = \{ 6 \}.
  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.
    Misc Q2(i)
    If x∈Ax \in A and A∈BA \in B, then x∈Bx \in B
  3. Misc Q2(ii)
    If A⊂BA \subset B and B∈CB \in C, then A∈CA \in C
  4. Misc Q2(iii)
    If A⊂BA \subset B and B⊂CB \subset C, then A⊂CA \subset C
  5. Misc Q2(iv)
    If A⊄BA \not\subset B and B⊄CB \not\subset C, then A⊄CA \not\subset C
  6. Misc Q2(v)
    If x∈Ax \in A and A⊄BA \not\subset B, then x∈Bx \in B
  7. Misc Q2(vi)
    If A⊂BA \subset B and x∉Bx \notin B, then x∉Ax \notin A
  8. Misc Q3
    Let AA, BB, and CC be the sets such that A∪B=A∪CA \cup B = A \cup C and A∩B=A∩CA \cap B = A \cap C. Show that B=CB = C.
  9. Show that the following four conditions are equivalent :
    Misc Q4(i)
    A⊂BA \subset B
  10. Misc Q4(ii)
    A−B=ϕA - B = \phi
  11. Misc Q4(iii)
    A∪B=BA \cup B = B
  12. Misc Q4(iv)
    A∩B=AA \cap B = A
  13. Misc Q5
    Show that if A⊂BA \subset B, then C−B⊂C−AC - B \subset C - A.
  14. Misc Q6
    Show that for any sets AA and BB, A=(A∩B)∪(A−B)A = (A \cap B) \cup (A - B) and A∪(B−A)=(A∪B)A \cup (B - A) = (A \cup B).
  15. Using properties of sets, show that
    Misc Q7(i)
    A∪(A∩B)=AA \cup (A \cap B) = A
  16. Misc Q7(ii)
    A∩(A∪B)=AA \cap (A \cup B) = A.
  17. Misc Q8
    Show that A∩B=A∩CA \cap B = A \cap C need not imply B=CB = C.
  18. Misc Q9
    Let AA and BB be sets. If A∩X=B∩X=ϕA \cap X = B \cap X = \phi and A∪X=B∪XA \cup X = B \cup X for some set XX, show that A=BA = B. (Hints A=A∩(A∪X)A = A \cap (A \cup X), B=B∩(B∪X)B = B \cap (B \cup X) and use Distributive law)
  19. Misc Q10
    Find sets AA, BB and CC such that A∩BA \cap B, B∩CB \cap C and A∩CA \cap C are non-empty sets and A∩B∩C=ϕA \cap B \cap C = \phi.