Mathematics · Textbook solutions

Sets and Relations

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 165 questions

5.1.2 Representation of a Set

4 q

Worked Examples

Worked · 4
  1. State the sets using Roster method.
    5.1.2 SolvedEx.1(i)
    BB is the set of all days in a week.
  2. 5.1.2 SolvedEx.1(ii)
    CC is the set of all vowels in English alphabets.
  3. State the sets using set-Builder method.
    5.1.2 SolvedEx.2(i)
    Y={Jan, Feb, Mar, Apr, ...., Dec}Y = \{\text{Jan, Feb, Mar, Apr, ...., Dec}\}
  4. 5.1.2 SolvedEx.2(ii)
    B={1,4,9,16,25,.....}B = \{1, 4, 9, 16, 25, .....\}

5.1.5 Operations on Sets

3 q

Worked Examples

Worked · 3
  1. 5.1.5 SolvedEx.1
    A={x/x is a prime number less than 10}A = \{x / x \text{ is a prime number less than } 10\}, B={x/x∈N, x is a factor of 8}B = \{x / x \in N,\ x \text{ is a factor of } 8\}. Find A∪BA \cup B.
  2. 5.1.5 SolvedEx.2
    A={1,3,5,7,9}A = \{1, 3, 5, 7, 9\} B={1,2,3,4,5,6,7,8}B = \{1, 2, 3, 4, 5, 6, 7, 8\} Find A∩BA \cap B.
  3. 5.1.5 SolvedEx.3
    A={1,3,5,7,9}A = \{1, 3, 5, 7, 9\} B={2,4,6,8,10}B = \{2, 4, 6, 8, 10\}, A∩B=?A \cap B = ?

5.1.6 Intervals

12 q

Worked Examples

Worked · 12
  1. Solve the following inequalities and write the solution set using interval notation.
    5.1.6 SolvedEx.1(a)
    −7<2x+5≤9-7 < 2x + 5 \le 9
  2. 5.1.6 SolvedEx.1(b)
    x2+2x<15x^2 + 2x < 15
  3. 5.1.6 SolvedEx.1(c)
    xx−3≥4\frac{x}{x-3} \ge 4
  4. If A=[−5,3]A = [-5, 3], B=(3,7)B = (3, 7), and (5,8)(5, 8). Find
    5.1.6 SolvedEx.2(a)
    A∪BA \cup B
  5. 5.1.6 SolvedEx.2(b)
    B∪CB \cup C
  6. 5.1.6 SolvedEx.2(c)
    A∪CA \cup C
  7. 5.1.6 SolvedEx.2(d)
    A∩BA \cap B
  8. 5.1.6 SolvedEx.2(e)
    B∩CB \cap C
  9. 5.1.6 SolvedEx.2(f)
    A∩CA \cap C
  10. 5.1.6 SolvedEx.2(g)
    A∪C′A \cup C'
  11. 5.1.6 SolvedEx.2(h)
    B−CB - C
  12. 5.1.6 SolvedEx.2(i)
    C−BC - B

5.1 Sets

10 q

Solved Examples

Worked · 10
  1. 5.1 SolvedEx.1
    If A={x/x∈N, x is a factor of 6}A = \{x / x \in \mathrm{N},\ x \text{ is a factor of } 6\}, B={x/x∈N, x is a factor of 8}B = \{x / x \in \mathrm{N},\ x \text{ is a factor of } 8\}, find the A−BA - B and B−AB - A.
  2. 5.1 SolvedEx.2
    A={1x/x∈N, x<8}A = \left\{\dfrac{1}{x} / x \in \mathrm{N},\ x < 8\right\}, B={12x/x∈N, x≤8}B = \left\{\dfrac{1}{2x} / x \in \mathrm{N},\ x \le 8\right\}. Find A−BA - B and B−AB - A.
  3. 5.1 SolvedEx.3
    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\}, D={7,8,9,10}D = \{7, 8, 9, 10\} find i) A∪BA \cup B ii) A∪B∪CA \cup B \cup C iii) B∪C∪DB \cup C \cup D. Are the sets A, B, C, D equivalent?
  4. 5.1 SolvedEx.4
    Let U={1,2,3,4,5,6,7,8,9,10}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} be the universal set, A={1,3,5,7,9}A = \{1, 3, 5, 7, 9\}, B={2,3,4,6,8,10}B = \{2, 3, 4, 6, 8, 10\}, C={6,7,8,9}C = \{6, 7, 8, 9\} find i) A′A' ii) (A∩C)′(A \cap C)' iii) (A′)′(A')' iv) (B−C)′(B - C)'
  5. 5.1 SolvedEx.5
    Let X be the universal set, for the non-empty sets A and B, verify the De Morgan's laws i) (A∪B)′=A′∩B′(A \cup B)' = A' \cap B' ii) (A∩B)′=A′∪B′(A \cap B)' = A' \cup B'. Where X={1,2,3,4,5,6,7,8,9,10}X = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, A={1,2,3,4,5}A = \{1, 2, 3, 4, 5\}, B={1,2,5,6,7}B = \{1, 2, 5, 6, 7\}
  6. 5.1 SolvedEx.6
    If P={x/x2+14x+40=0}P = \{x / x^2 + 14x + 40 = 0\}, Q={x/x2−5x+6=0}Q = \{x / x^2 - 5x + 6 = 0\}, R={x/x2+17x−60=0}R = \{x / x^2 + 17x - 60 = 0\} and the universal set X={−20,−10−4,2,3,4}X = \{-20, -10 - 4, 2, 3, 4\}, find i) P∪QP \cup Q ii) Q∩RQ \cap R iii) P∪(Q∩R)P \cup (Q \cap R) iv) P∩(Q∪R)P \cap (Q \cup R) v) (P∪Q)′(P \cup Q)' vi) Q′∩R′Q' \cap R'
  7. 5.1 SolvedEx.7
    If A and B are the subsets of the universal set X and n(x)=50n(x) = 50, n(A)=35n(A) = 35, n(B)=22n(B) = 22 and n(A′∩B′)=3n(A' \cap B') = 3, find i) n(A∪B)n(A \cup B) ii) n(A∩B)n(A \cap B) iii) n(A′∩B)n(A' \cap B) iv) n(A∪B′)n(A \cup B')
  8. 5.1 SolvedEx.8
    In a board examination, 40 students failed in Physics, 40 in Chemistry and 35 in Maths, 20 failed in Maths and Physics, 17 in Physics and Chemistry, 15 in Maths and Chemistry and 5 in all the three subjects. If 350 students appeared in the examination, how many of them did not fail in any subject?
  9. 5.1 SolvedEx.9
    A company produces three kinds of products A, B and C. The company studied the perference of 1600 consumers for these 3 products. It was found that the product A was liked by 1250, the product B was liked by 930 and product C was liked by 1000. The proudcts A and B were liked by 650, the products B and C were liked by 610 and the products C and A were liked by 700 consumers. None of the products were liked by 30 consumers. Find number of consumers who liked. i) All the three products ii) Only two of these products.
  10. 5.1 SolvedEx.10
    In a survey of 100 consumers 72 like product A and 45 like product B. Find the least and the most number that must have liked both products A and B.

Exercise 5.1

55 q
  1. Describe the following sets in Roster form
    Ex 5.1 Q1(i)
    A={x/x is a letter of the word ’MOVEMENT’}A = \{x/x \text{ is a letter of the word 'MOVEMENT'}\}
  2. Ex 5.1 Q1(ii)
    B={x/x is an integer, −32<x<92}B = \left\{x/x \text{ is an integer},\ -\frac{3}{2} < x < \frac{9}{2}\right\}
  3. Ex 5.1 Q1(iii)
    C={x/x=2n+1, n∈N}C = \{x/x = 2n + 1,\ n \in N\}
  4. Describe the following sets in Set-Builder form
    Ex 5.1 Q2(i)
    {0}\{0\}
  5. Ex 5.1 Q2(ii)
    (0,±1,±2,±3}(0, \pm 1, \pm 2, \pm 3\}
  6. Ex 5.1 Q2(iii)
    {12,25,310,417,526,637,750}\left\{\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \frac{5}{26}, \frac{6}{37}, \frac{7}{50}\right\}
  7. Ex 5.1 Q2(iv)
    {0,−1,2,−3,4,−5,6,…}\{0, -1, 2, -3, 4, -5, 6, \ldots\}
  8. If A={x/6x2+x−15=0}A = \{x/6x^2 + x - 15 = 0\}, B={x/2x2−5x−3=0}B = \{x/2x^2 - 5x - 3 = 0\}, C={x/2x2−x−3=0}C = \{x/2x^2 - x - 3 = 0\} then find
    Ex 5.1 Q3(i)
    (A∪B∪C)(A \cup B \cup C)
  9. Ex 5.1 Q3(ii)
    (A∩B∩C)(A \cap B \cap C)
  10. Ex 5.1 Q4
    If A, B, C are the sets for the letters in the words 'college', 'marriage' and 'luggage' respective, then verify that [A−(B∪C)]=[(A−B)∩(A−C)][A - (B \cup C)] = [(A - B) \cap (A - C)]
  11. If A={1,2,3,4}A = \{1, 2, 3, 4\}, B={3,4,5,6}B = \{3, 4, 5, 6\}, C={4,5,6,7,8}C = \{4, 5, 6, 7, 8\} and universal set X={1,2,3,4,5,6,7,8,9,10}X = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, then verify the following:
    Ex 5.1 Q5(i)
    A∪(B∩C)=(A∪B)∩(A∪C)A \cup (B \cap C) = (A \cup B) \cap (A \cup C)
  12. Ex 5.1 Q5(ii)
    A∩(B∪C)=(A∩B)∪(A∪C)A \cap (B \cup C) = (A \cap B) \cup (A \cup C)
  13. Ex 5.1 Q5(iii)
    (A∪B)′=(A′∩B)′(A \cup B)' = (A' \cap B)'
  14. Ex 5.1 Q5(iv)
    (A∩B)′=A′∪B′(A \cap B)' = A' \cup B'
  15. Ex 5.1 Q5(v)
    A=(A∩B)∪(A∩B′)A = (A \cap B) \cup (A \cap B')
  16. Ex 5.1 Q5(vi)
    B=(A∩B)∪(A′∩B)B = (A \cap B) \cup (A' \cap B)
  17. Ex 5.1 Q5(vii)
    (A∪B)=(A−B)∪(A∩B)∪(B−A)(A \cup B) = (A - B) \cup (A \cap B) \cup (B - A)
  18. Ex 5.1 Q5(viii)
    A∩(B Δ C)=(A∩B) Δ (A∩C)A \cap (B\, \Delta\, C) = (A \cap B)\, \Delta\, (A \cap C)
  19. Ex 5.1 Q5(ix)
    n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)
  20. Ex 5.1 Q5(x)
    n(B)=(A′∩B)+n(A∩B)n(B) = (A' \cap B) + n(A \cap B)
  21. If A and B are subsets of the universal set X and n(X)=50n(X) = 50, n(A)=35n(A) = 35, n(B)=20n(B) = 20, n(A′∩B′)=5n(A' \cap B') = 5, find
    Ex 5.1 Q6(i)
    n(A∪B)n(A \cup B)
  22. Ex 5.1 Q6(ii)
    n(A∩B)n(A \cap B)
  23. Ex 5.1 Q6(iii)
    n(A′∩B)n(A' \cap B)
  24. Ex 5.1 Q6(iv)
    n(A∩B′)n(A \cap B')
  25. In a class of 200 students who appeared certain examinations, 35 students failed in CET, 40 in NEET and 40 in JEE, 20 failed in CET and NEET, 17 in NEET and JEE, 15 in CET and JEE and 5 failed in all three examinations. Find how many students,
    Ex 5.1 Q7(i)
    did not fail in any examination.
  26. Ex 5.1 Q7(ii)
    failed in NEET or JEE entrance.
  27. From amongst 2000 literate individuals of a town, 70% read Marathi newspapers, 50% read English newspapers and 32.5% read both Marathi and English newspapers. Find the number of individuals who read.
    Ex 5.1 Q8(i)
    at least one of the newspapers.
  28. Ex 5.1 Q8(ii)
    neither Marathi and English newspaper.
  29. Ex 5.1 Q8(iii)
    Only one of the newspapers.
  30. Ex 5.1 Q9
    In a hostel, 25 students take tea, 20 students take coffee, 15 students take milk, 10 student take bot tea and coffee, 8 students take both milk and coffee. None of them take tea and milk both and everyone takes atleast one beverage, find the total number of students in the hostel.
  31. There are 260 persons with a skin disorder. If 150 had been exposed to the chemical A, 74 to the chemical B, and 36 to both chemicals A and B, find the number of persons exposed to
    Ex 5.1 Q10(i)
    Chemical A but not Chemical B
  32. Ex 5.1 Q10(ii)
    Chemical B but not Chemical A
  33. Ex 5.1 Q10(iii)
    Chemical A or Chemical B
  34. Ex 5.1 Q11
    Write down the power set of A={1,2,3}A = \{1, 2, 3\}
  35. Write the following intervals in Set-Builder form
    Ex 5.1 Q12(i)
    (−3,0)(-3, 0)
  36. Ex 5.1 Q12(ii)
    [6,12][6, 12]
  37. Ex 5.1 Q12(iii)
    (6,∞)(6, \infty)
  38. Ex 5.1 Q12(iv)
    (−∞,5](-\infty, 5]
  39. Ex 5.1 Q12(v)
    (2,5](2, 5]
  40. Ex 5.1 Q12(vi)
    [−3,4)[-3, 4)
  41. Ex 5.1 Q13
    A college awarded 38 medals in volley ball, 15 in football and 20 in basket ball. The medals awarded to a total of 58 players and only 3 players got medals in all three sports. How many received medals in exactly two of the three sports?
  42. Solve the following inequalities and write the solution set using interval notation.
    Ex 5.1 Q14(i)
    −9<2x+7≤19-9 < 2x + 7 \le 19
  43. Ex 5.1 Q14(ii)
    x2−x>20x^2 - x > 20
  44. Ex 5.1 Q14(iii)
    2xx−4≤5\frac{2x}{x - 4} \le 5
  45. Ex 5.1 Q14(iv)
    6x2+1≤5x6x^2 + 1 \le 5x
  46. If A=(−7,3]A = (-7, 3], B=[2,6]B = [2, 6] and C=[4,9]C = [4, 9] then find
    Ex 5.1 Q15(i)
    A∪BA \cup B
  47. Ex 5.1 Q15(ii)
    B∪CB \cup C
  48. Ex 5.1 Q15(iii)
    A∪CA \cup C
  49. Ex 5.1 Q15(iv)
    A∩BA \cap B
  50. Ex 5.1 Q15(v)
    B∩CB \cap C
  51. Ex 5.1 Q15(vi)
    A∩CA \cap C
  52. Ex 5.1 Q15(vii)
    A′∩BA' \cap B
  53. Ex 5.1 Q15(viii)
    B′∩C′B' \cap C'
  54. Ex 5.1 Q15(ix)
    B−CB - C
  55. Ex 5.1 Q15(x)
    A−BA - B

5.2.1 Ordered Pair

1 q

Worked Examples

Worked · 1
  1. 5.2.1 SolvedEx.1
    Find xx and yy when (x+3, 2)=(4, y−3)(x + 3,\ 2) = (4,\ y - 3).

5.2.2 Cartesian Product

1 q

Worked Examples

Worked · 1
  1. 5.2.2 SolvedEx.1
    Let A={1,3}A = \{1, 3\}; B={2,3,4}B = \{2, 3, 4\}. Find the number of elements in the Cartesian product of AA and BB.

5.2.4 Domain, Co-domain and Range

1 q

Worked Examples

Worked · 1
  1. 5.2.4 SolvedEx.1
    Let A={1,2,3,4,5}A = \{1, 2, 3, 4, 5\} and B={1,4,5}B = \{1, 4, 5\}. Let RR be a relation such that (x,y)∈R(x, y) \in R implies x<yx < y. List the elements of RR.

5.2.5 Relations — Illustrative Examples

4 q

Worked Examples

Worked · 4
  1. 5.2.5 SolvedEx.1
    Let A={1,2,3}A = \{1, 2, 3\} and R={(1,2),(2,2),(3,1),(3,2)}R = \{(1, 2), (2, 2), (3, 1), (3, 2)\}. Show that RR is a binary relation on AA.
  2. 5.2.5 SolvedEx.2
    Let NN be the set of all natural numbers and R={(a,b)/a,b∈N and 2a+b=10}R = \{(a, b) / a, b \in N \text{ and } 2a + b = 10\}. Show that RR is a binary relation on NN, list its elements, and state its domain, co-domain and range.
  3. 5.2.5 SolvedEx.3
    R′R' is defined in Z×ZZ \times Z as aR′baR'b if (a+b)(a + b) is even. Let RR be defined in Z×ZZ \times Z as aRbaRb if (a−b)(a - b) is even. Show that RR and R′R' define the same relation, and describe the subset of Z×ZZ \times Z related to them.
  4. 5.2.5 SolvedEx.4
    If A={2,4,6}A = \{2, 4, 6\}, write R=A×AR = A \times A and state what type of relation it is on AA.

5.2.7 Types of Relations

3 q

Worked Examples

Worked · 3
  1. 5.2.7 SolvedEx.1
    Let RR be a relation on QQ, defined by R={(a,b)/a,b∈Q and a−b∈Z}R = \{(a, b) / a, b \in Q \text{ and } a - b \in Z\}. Show that RR is an equivalence relation.
  2. 5.2.7 SolvedEx.2
    Show that the relation "is congruent to" on the set of all triangles in a plane is an equivalence relation.
  3. 5.2.7 SolvedEx.3
    Show that R={(a,b):a,b∈Z, a≡b (mod m)}R = \{(a, b) : a, b \in Z,\ a \equiv b \ (\mathrm{mod}\ m)\} is an equivalence relation.

5.2 Relations

8 q

Solved Examples

Worked · 8
  1. 5.2 SolvedEx.1
    If (x+1,y−2)=(3,1)(x+1, y-2) = (3, 1) find the value of xx and yy.
  2. 5.2 SolvedEx.2
    If A={1,2}A = \{1, 2\}, find A×AA \times A.
  3. 5.2 SolvedEx.3
    If A={1,3,5}A = \{1, 3, 5\} and B={2,3}B = \{2, 3\} find A×BA \times B and B×AB \times A show that A×B≠B×AA \times B \ne B \times A.
  4. 5.2 SolvedEx.5
    A and B are two sets given in such a way that A×BA \times B contains 6 elements. If three elements of A×BA \times B are (1,3)(1, 3), (2,5)(2, 5) and (3,3)(3, 3), find its remaining elements.
  5. 5.2 SolvedEx.6
    Express {(x,y)/x2+y2=25\{(x, y) / x^2 + y^2 = 25 where x,y∈W}x, y \in W\} as a set of ordered pairs.
  6. 5.2 SolvedEx.7
    Let A={1,2,3}A = \{1, 2, 3\} and B={2,4,6}B = \{2, 4, 6\} Show that R={(1,2),(1,4),(3,2),(3,4)}R = \{(1, 2), (1, 4), (3, 2), (3, 4)\} is a relation from A to B. Find i) domain (R) ii) Co-domain (R) iii) Range (R).
  7. 5.2 SolvedEx.8
    Let A={1,2,3,4,5}A = \{1, 2, 3, 4, 5\} and B={1,4,5}B = \{1, 4, 5\}. Let R be a relation from A to B. Such that (x,y)∈R(x, y) \in R if 2x<y2x < y. i) List the elements of R. ii) Find the domain, co-domain and range of R.
  8. 5.2 SolvedEx.9
    Let A={1,2,3,4,5,6}A = \{1, 2, 3, 4, 5, 6\} Define a relation R from A to A by R={(x,y)/y=x+1}R = \{(x, y) / y = x + 1\}. Write down the domain, co-domain and range of R.

Exercise 5.2

24 q
  1. Ex 5.2 Q1
    If (x−1,y+4)=(1,2)(x-1, y+4) = (1, 2) find the values of xx and yy.
  2. Ex 5.2 Q2
    If (x+13,y3−1)=(12,32)\left(x + \frac{1}{3}, \frac{y}{3} - 1\right) = \left(\frac{1}{2}, \frac{3}{2}\right), find xx and yy.
  3. Ex 5.2 Q3
    If A={a,b,c}A = \{a, b, c\}, B=(x,y)B = (x, y) find A×BA \times B, B×AB \times A, A×AA \times A, B×BB \times B.
  4. Ex 5.2 Q4
    If P={1,2,3)P = \{1, 2, 3) and Q={1,4}Q = \{1, 4\}, find sets P×QP \times Q and Q×PQ \times P.
  5. Let A={1,2,3,4)A = \{1, 2, 3, 4), B={4,5,6}B = \{4, 5, 6\}, C={5,6}C = \{5, 6\}. Verify,
    Ex 5.2 Q5(i)
    A×(B∩C)=(A×B)∩(A×C)A \times (B \cap C) = (A \times B) \cap (A \times C)
  6. Ex 5.2 Q5(ii)
    A×(B∪C)=(A×B)∪(A×C)A \times (B \cup C) = (A \times B) \cup (A \times C)
  7. Ex 5.2 Q6
    Express {(x,y)/x2+y2=100,\{(x, y) / x^2 + y^2 = 100, where x,y∈W}x, y \in W\} as a set of ordered pairs.
  8. Ex 5.2 Q7
    Let A={6,8}A = \{6, 8\} and B={1,3,5}B = \{1, 3, 5\} Show that R1={(a,b)/a∈A,b∈B,a−bR_1 = \{(a, b) / a \in A, b \in B, a - b is an even number}\} is a null relation. R2={(a,b)/a∈A,b∈B,a+bR_2 = \{(a, b) / a \in A, b \in B, a + b is odd number}\} is an universal relation.
  9. Write the relation in the Roster form. State its domain and range.
    Ex 5.2 Q8(i)
    R1={(a,a2)/aR_1 = \{(a, a^2) / a is prime number less than 15}15\}
  10. Ex 5.2 Q8(ii)
    R2={(a,1a)/0<a≤5,a∈N}R_2 = \left\{\left(a, \frac{1}{a}\right) / 0 < a \le 5, a \in N\right\}
  11. Ex 5.2 Q8(iii)
    R3={(x,y)/y=3x,y∈{3,6,9,12},x∈{1,2,3}}R_3 = \{(x, y) / y = 3x, y \in \{3, 6, 9, 12\}, x \in \{1, 2, 3\}\}
  12. Ex 5.2 Q8(iv)
    R4={(x,y)/y>x+1,x=1,2R_4 = \{(x, y) / y > x + 1, x = 1, 2 and y=2,4,6}y = 2, 4, 6\}
  13. Ex 5.2 Q8(v)
    R5={(x,y)/x+y=3,x,y∈{0,1,2,3}}R_5 = \{(x, y) / x + y = 3, x, y \in \{0, 1, 2, 3\}\}
  14. Ex 5.2 Q8(vi)
    R6={(a,b)/a∈N,a<6R_6 = \{(a, b) / a \in N, a < 6 and b=4}b = 4\}
  15. Ex 5.2 Q8(vii)
    R7={(a,b)/a,b∈N,a+b=6}R_7 = \{(a, b) / a, b \in N, a + b = 6\}
  16. Ex 5.2 Q8(viii)
    R8={(a,b)/b=a+2,a∈z,0<a<5}R_8 = \{(a, b) / b = a + 2, a \in z, 0 < a < 5\}
  17. Identify which of the following relations are reflexive, symmetric and transitive. For each, state whether it has each of the three properties and justify briefly.
    Ex 5.2 Q9(i)
    R={(a,b):a,b∈Z, a−b is an integer}R = \{(a, b) : a, b \in \mathbb{Z},\ a - b \text{ is an integer}\}
  18. Ex 5.2 Q9(ii)
    R={(a,b):a,b∈N, a+b is even}R = \{(a, b) : a, b \in \mathbb{N},\ a + b \text{ is even}\}
  19. Ex 5.2 Q9(iii)
    R={(a,b):a,b∈N, a divides b}R = \{(a, b) : a, b \in \mathbb{N},\ a \text{ divides } b\}
  20. Ex 5.2 Q9(iv)
    R={(a,b):a,b∈N, a2−4ab+3b2=0}R = \{(a, b) : a, b \in \mathbb{N},\ a^2 - 4ab + 3b^2 = 0\}
  21. Ex 5.2 Q9(v)
    R={(a,b):a is sister of b, a,b∈G=set of girls}R = \{(a, b) : a \text{ is sister of } b,\ a, b \in G = \text{set of girls}\}
  22. Ex 5.2 Q9(vi)
    R={(a,b):line a is perpendicular to line b in a plane}R = \{(a, b) : \text{line } a \text{ is perpendicular to line } b \text{ in a plane}\}
  23. Ex 5.2 Q9(vii)
    R={(a,b):a,b∈R, a<b}R = \{(a, b) : a, b \in \mathbb{R},\ a < b\}
  24. Ex 5.2 Q9(viii)
    R={(a,b):a,b∈R, a≤b3}R = \{(a, b) : a, b \in \mathbb{R},\ a \le b^3\}

Miscellaneous Exercise 5

39 q

(I) Select the correct option

Practice · 10
  1. Misc I Q1
    For the set A={a,b,c,d,e}A = \{a, b, c, d, e\} the correct statement is
    1. A.
      {a,b}∈A\{a, b\} \in A
    2. B.
      {a}∈A\{a\} \in A
    3. C.
      a∈Aa \in A
    4. D.
      a∉Aa \notin A
  2. Misc I Q2
    If aN={ax:x∈N}aN = \{ax : x \in N\}, then set 6N∩8N=6N \cap 8N =
    1. A.
      8N8N
    2. B.
      48N48N
    3. C.
      12N12N
    4. D.
      24N24N
  3. Misc I Q3
    If set A is empty set then n[P[P[P(A)]]]n[P[P[P(A)]]] is
    1. A.
      6
    2. B.
      16
    3. C.
      2
    4. D.
      4
  4. Misc I Q4
    In a city 20% of the population travels by car, 50% travels by bus and 10% travels by both car and bus. Then, persons travelling by car or bus are
    1. A.
      80%
    2. B.
      40%
    3. C.
      60%
    4. D.
      70%
  5. Misc I Q5
    If the two sets A and B are having 43 elements in common, then the number of elements common to each of the sets A×BA \times B and B×AB \times A is
    1. A.
      43243^2
    2. B.
      2432^{43}
    3. C.
      434343^{43}
    4. D.
      2862^{86}
  6. Misc I Q6
    Let R be a relation on the set N be defined by {(x,y)/x,y∈N,2x+y=41}\{(x, y) / x, y \in N, 2x + y = 41\} Then R is
    1. A.
      Reflexive
    2. B.
      Symmetric
    3. C.
      Transitive
    4. D.
      None of these
  7. Misc I Q7
    The relation ">" in the set of N (Natural number) is
    1. A.
      Symmetric
    2. B.
      Reflexive
    3. C.
      Transitive
    4. D.
      Equivalent relation
  8. Misc I Q8
    A relation between A and B is
    1. A.
      only A×BA \times B
    2. B.
      An Universal set of A×BA \times B
    3. C.
      An equivalent set of A×BA \times B
    4. D.
      A subset of A×BA \times B
  9. Misc I Q9
    If (x,y)∈N×N(x, y) \in N \times N, then xy=x2xy = x^2 is a relation which is
    1. A.
      Symmetric
    2. B.
      Reflexive
    3. C.
      Transitive
    4. D.
      Equivalence
  10. Misc I Q10
    If A={a,b,c}A = \{a, b, c\} The total no. of distinct relations in A×AA \times A is
    1. A.
      3
    2. B.
      9
    3. C.
      8
    4. D.
      292^9

(II)

Practice · 29
  1. Write down the following sets in set builder form
    Misc II Q1(i)
    {10,20,30,40,50}\{10, 20, 30, 40, 50\}
  2. Misc II Q1(ii)
    {a,e,i,o,u)\{a, e, i, o, u)
  3. Misc II Q1(iii)
    {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}\{\text{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}\}
  4. If U={x/x∈N,1≤x≤12}U = \{x / x \in N, 1 \le x \le 12\}, A={1,4,7,10}A = \{1, 4, 7, 10\}, B={2,4,6,7,11}B = \{2, 4, 6, 7, 11\}, C={3,5,8 9,12}C = \{3, 5, 8\ 9, 12\}, write down the sets
    Misc II Q2(i)
    A∪BA \cup B
  5. Misc II Q2(ii)
    B∩CB \cap C
  6. Misc II Q2(iii)
    A−BA - B
  7. Misc II Q2(iv)
    B∩C′B \cap C'
  8. Misc II Q2(v)
    A∪B∪CA \cup B \cup C
  9. Misc II Q2(vi)
    A∩(B∪C)A \cap (B \cup C)
  10. Misc II Q3
    In a survey of 425 students in a school, it was found that 115 drink apple juice, 160 drink orange juice and 80 drink both apple as well as orange juice. How many drink neither apple juice nor orange juice?
  11. Misc II Q4
    In a school there are 20 teachers who teach Mathematics or Physics. Of these, 12 teach Mathematics and 4 teach both Physics and Mathematics. How many teachers teach Physics?
  12. Misc II Q5(i)
    If A={1,2,3}A = \{1, 2, 3\} and B={2,4}B = \{2, 4\}, state the elements of A×AA \times A, A×BA \times B, B×AB \times A, B×BB \times B, (A×B)∩(B×A)(A \times B) \cap (B \times A)
  13. Misc II Q5(ii)
    If A={−1,1}A = \{-1, 1\}, find A×A×AA \times A \times A
  14. If A={1,2,3}A = \{1, 2, 3\}, B={4,5,6}B = \{4, 5, 6\} check if the following are relations from A to B. Also write its domain and range.
    Misc II Q6(i)
    R1={(1,4),(1,5),(1,6)}R_1 = \{(1, 4), (1, 5), (1, 6)\}
  15. Misc II Q6(ii)
    R2={(1,5),(2,4),(3,6)}R_2 = \{(1, 5), (2, 4), (3, 6)\}
  16. Misc II Q6(iii)
    R3={(1,4),(1,5),(3,6),(2,6),(3,4)}R_3 = \{(1, 4), (1, 5), (3, 6), (2, 6), (3, 4)\}
  17. Misc II Q6(iv)
    R4={(4,2),(2,6),(5,1),(2,4)}R_4 = \{(4, 2), (2, 6), (5, 1), (2, 4)\}
  18. Determine the domain and range of the following relations.
    Misc II Q7(i)
    R={(a,b)/a∈N,a<5,b=4}R = \{(a, b) / a \in N, a < 5, b = 4\}
  19. Misc II Q7(ii)
    R={(a,b)/b=∣a−1∣,a∈z,∣a∣<3}R = \{(a, b) / b = |a - 1|, a \in z, |a| < 3\}
  20. Find R:A→AR : A \to A when A={1,2,3,4}A = \{1, 2, 3, 4\} such that
    Misc II Q8(i)
    R=(a,b)/a−b=10}R = (a, b) / a - b = 10\}
  21. Misc II Q8(ii)
    R={(a,b)/∣a−b∣)≥0}R = \{(a, b) / |a - b|) \ge 0\}
  22. R={1,2,3}→{1,2,3}R = \{1, 2, 3\} \to \{1, 2, 3\} given by R={(1,1)(2,2),(3,3),(1,2)(2,3)}R = \{(1,1) (2,2), (3,3), (1,2) (2,3)\} Check if R is
    Misc II Q9(a)
    relflexive
  23. Misc II Q9(b)
    symmentric
  24. Misc II Q9(c)
    transitive
  25. Misc II Q10
    Check if R:Z→ZR : Z \to Z, R={(a,b)∣2 divides a−b}R = \{(a, b) | 2 \text{ divides } a - b\} is equivalence relation.
  26. Misc II Q11
    Show that the relation R in the set A={1,2,3,4,5}A = \{1, 2, 3, 4, 5\} Given by R={(a,b)/∣a−b∣ is even}R = \{(a, b) / |a - b| \text{ is even}\} is an equivalence relation
  27. Show that following are equivalence relation
    Misc II Q12(a)
    R in A is set of all books. given by R={(x,y)/x and y have same number of pages}R = \{(x, y) / x \text{ and } y \text{ have same number of pages}\}
  28. Misc II Q12(b)
    R in A={x∈Z∣0≤x≤12}A = \{x \in Z | 0 \le x \le 12\} given by R={(a,b)/∣a−b∣ is a multiple of 4}R = \{(a, b) / |a - b| \text{ is a multiple of } 4\}
  29. Misc II Q12(c)
    R in A={x∈N/x≤10}A = \{x \in N / x \le 10\} given by R={(a,b)∣a=b}R = \{(a, b) | a = b\}