Set, Relations and Functions Exercise

Page # 10 SET, RELATIONS AND FUNCTIONS EXERCISE Q.1 Q.2 Q.3 Q.10 State in each whether A Rewrite the following statements, using set notations : (i) A is an empty set and B is a non-empty set. (ii) number of elements in A is 6. (iii)0 is a whole number but not a natural number. Q.11 Write down all subsets of each of the following (i) A = {3} (ii) B = {-2, 5} Let A = [2, 4, 6, 8, 10]. Insert the appropriate symbol  or  in the blank spaces : (i) 6 ……..A (ii) 1 …….. A (iii) 5 …….. A (iv) 10 ….. A (v) 2 ………. A (vi) 3 ……… A Q.4 Q.5 Write the following sets in the roster form (i) A = set of all letters in the word ‘LETTER’ (ii) B ={x: x is a natural number less then 7} (iii) C = {x : x is an integer and 2  x < 5} List all the elements of each of the following sets : 1 7  (i) A = x : x  Zand   x   2 2   Z and x2  4} (iii) C = {x : x = 2n, n  N and n  5} (ii) B = {x : x Q.6 State whether the give set is finite or infinite: Q.12 If A = {1, 2, 3}, find P(A) and n{P(A)}. Q.13 Express each of the following sets as an interval  R, 4 < x < 0} (ii) B = {x : x  R, 0  x < 3} (iii) C = {x : x  R, 2 < x  6} (i) A = {x : x Q.14 Write each of the following intervals in the setbuilder from: (i) A = ]-2, 2[ (ii) B = [5, 9] (iii) C = [8, 11] Q.15 If A = {3, {4, 5}, 6}, find which of the following statement are true: (i) {4, 5} A (ii) {4, 5} A (iii){{4, 5}} A Q.16 If A = {a, b, c, d, e, f} B = {c, e, g, h} and C = {a, e, m, n}, find : (i) A  B (ii) B  C (iii) A  C Q.17 If A = (1, 2, 3, 4, 5}, B = (4, 5, 6, 7, 8}, C = {7,8,9,10,11} and D = {10,11,12,13,14}, find : (i) A  B (ii) B  C (iii) A  C (iv) B  D Which of the following sets are empty sets Q.19 If A = {x : x  N}, B = {x : x  N and x is even},  W, x + 3 < 3} (ii) B = {x : x  Q, 1 < x < 2} (iii) C = {0} Which of the following in the pairs of equal sets? (i) A = set of the letters in the word ‘ALLOY’ B = set of the letters in the word ‘LOYAL’  Z, x2  4} and D = {x : x  Z, x2  4 = 0} (iii) E = {x : x Z, x  3 = 0} and F = (x : x  Z, x2  9 = 0} (ii) C = {x : x Q.9 (iii) A = {x : x  Z, x2 = 1}, B = {x :x  N,x2=1} Q.18 If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}, find : (i) A = {x : x Q.8 (i) A = {0, 1, 2, 3}, B = {1, 2, 3, 4, 5} (ii) A = {x, y, z}, B = {z, y, x}  R; 0 < x < 1} (ii) B = {x  Z; x < 1} (iii) C = {x  Z; 15 < x < 15} (i) A = {x Q.7  B or A  B Which of the following collections are sets (i) The collection of all good athletes of India. (ii) The collection of fat boys of your locality. (iii) The collection of all those students of your class whose age exceeds 15 years. Which of the following are pairs of equivalent sets ? (i) A = {1, 2, 3}, B = {2, 4, 6} (ii) P = {0}, Q =  (iii) C = {2, 1, 0}, D = {1, 2, 3} (i) A B (ii) A C (iii) B C  N and x is odd} and D = {x : x  N and x is prime} then find: (i) A  B (ii) A  C (iii) A  D C = {x : x  N and 1  x < 4}, B = {(x + 2) : x  N and 2  x < 5} and C = {x : x  N and 4 < x < 8}, find: (i) A  B (ii) A  B (iii) (A  B)  C Q.20 If A = {2x : x Q.21 If A = {2, 4, 6, 8, 10, 12} and B = {3, 4, 5, 6, 7, 8, 10}, find: (i) (AB) (ii) (BA) (iii) (AB)  (BA) Q.22 If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}, find: (i) A  (BC) (ii) A(B  C) (iii) A(B  C) Page # 11 SET, RELATIONS AND FUNCTIONS 1   : x  N and x  8 and x  1   : x  N and x  4  , find: B=  2 x   (i) A  B (ii) A  B (iii) A–B Q.34 If A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A ={2,4,6,8} and B ={2, 3, 5, 7}, verify that: Q.23 If A =  B)’ = (A’  B’) (ii) (A  B)’ = (A’  B’) (i) (A (iv) B–A Q.24 If R is the set of all real numbers and Q is the set of all rational numbers then what is the set (R  Q) ? Q.25 If A = {2, 3, 5, 7, 11} and B = , find : (i) A B (ii) A B Q.26 If A and B are two sets such that A find: (i) A  B (ii) A  B  B, then (iii) A – B Q.27 Which of the following sets are pairs of disjoint sets: Justify your answer. (i) A = {3, 4, 5, 6} and B = {2, 5, 7, 9} (ii) C = {1, 2, 3, 4, 5} and D = (6, 7, 9, 11}  N, x is even and x < 8} F = {x : x = 3n, n  N and n < 4} (iii) E = {x : x Q.28 If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, S = (1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {1, 4, 5, 6}, find: (i) A’ (ii) B’ (iii) C’ (iv) (B’)’ (v) (A  B)’ (vi) (A  C)’ Q.29 If U = {a, b, c, d, e}, A = {a, b, c} and B = {b, c, d, e}, then verify that : (i) (A  B)’ = (A’  B’) (ii) (A  B)’ = (A’  B’) Q.30 If U is the universal set and A the blanks:  A’ = …. (iii) ’  A = ….. (i) A  U, then fill in  A’ = ….. (iv) U’  A = ….. (ii) A Q.31 If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}, verify that: (i) A  B = B  A (ii) A  C = C  A  C = C  B (iv) A  B = B  A (iii) B Q.32 If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}, verify that:  (B – C) = (A  B) – (A  C) (ii) A –(B  C) = (A – B)  (A – C) (i) A Q.33 If A = {x : xN, x  7}, B = {x: x is prime, x < 8} and C = {x : x  N, x is odd and x < 10}, verify that:  (B  C) = (A  B)  (A  C) (ii) A  (B  C) = (A  B)  (A  B) (i) A Q.35 Let A = {a, b, c}, B = {b, c, d, e} and C = {c, d, e, f} be subsets of U = {a, b, c, d, e, f}. Then, verify that:  B)’ = (A’  B’) (iii) (A  B)’ = (A’  B’) (i) (A’)’ = A (ii) (A Q.36 Given an example of three sets A, B, C such  B  , B  C  , A  C   and A  B  C = . that A Q.37 Let A = {a, b, c, e, f}, B = {c, d, e, g} and C = {b, c, f, g} be subsets of the set U = {a, b, c, d, e, f, g, h}. Draw Venn diagrams to represent the following sets: (i) A  B (ii) A  (B  C) (iii) A–B Q.38 Let A = {2, 4, 6, 8, 10}, B = {4, 8, 12, 16} and C = {6, 12, 18, 24}. Using Venn diagrams, verify that: A  (B  C) = (A  B)  (A  C) Q.39 Let A = {a, e, I, o, u}, B = {a, d, e, o, v} and C = {e, o, t, m}. Using Venn diagrams, verify that: A  (B  C) = (A  B)  (A  C) Q.40 Let A B U. Exhibit it in a Venn diagram. Q.41 Let A = {2, 3, 5, 7, 11, 13} and B = {5, 7, 9, 11, 15} be subsets of U = {2, 3, 5, 7, 9, 11, 13, 15}. Using Venn diagrams, verify that: (i) (A  B)’ = (A’  B’) (ii) (A  B)’ = (A’  B’) Q.42 Using Venn diagrams, show that (A–B),(AB) and (B – A) are disjoint sets, taking A = {2,4,6,8,10,12} and B = {3,6,9,12,15}. Q.43 If A and B are two sets such that n(A)=37, n(B)=26 and n(A  B)= 51, find n(A  B). Q.44 If P and Q are two sets such that n(P  Q) = 75, n(P  Q) = 17 and n(P) = 49, find n(Q). Q.45 If A and B are two sets such that n(A) = 24,  B) = 8, find: (ii) n(A  B) (iii) n(B  A) n(B) = 22 and n(A (i) n(A  B) Q.46 If A and B are two sets such that n(A–B)=24, n(B–A)=19 and n(A  B) = 11, find: (i) n(A) (ii) n(B) (iii) n(A  B) Page # 12 SET, RELATIONS AND FUNCTIONS Q.47 In a committee, 50 people speak Hindi, 20 speak English and 10 speak both Hindi and English. How many speak at least one of these two languages ? Q.48 In a group of 50 persons, 30 like tea, 25 like coffee and 16 like both. How many like (i) either tea or coffee? (ii) neither tea nor coffee? Q.49 In a class of a certain school 50 students offered mathematics, 42 offered biology and 24 offered both the subjects. Find the number of students. (i) offering mathematics only, (ii) offering biology only and (iii) offering any of the two subjects. Q.50 In an examination, 56% of the candidates failed in English and 48% failed in science. If 18% failed in both English and Sicience, find the percentage of those who passed in both the subjects. Q.51 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 ? Q.52 A school awarded 42 medals in hockey, 18 in basketball and 23 in cricket. If these medals were bagged by a total of 65 students and only (iii) how many are taking exactly one of the three subjects. Q.55 In a survey of 100 students, the number of students studying the various languages is found as: English only 18; English but not Hindi 23; English and Sanskrit 8; Sanskrit and Hindi 8; English 26; Sanskrit 48 and no language 24. Find (i) how many students are studying Hindi; (ii) how many students are studying English and Hindi both. Q.56 Find the values of a and b, when: (i) (a + 3, b (ii) (a + b, 2b Q.58 If P={a,b}and Q={x,y,z}, show that P×QQ×P. Q.59 If A = {2, 3, 5} and B = {5, 7}, find (i) A × B (ii) B × A (iii) A × A Q.60 If A = {x find (A × B) and (B × A). Is (A × B) = (B × A)? Q.61 If A={1,3,5}, B={3,4}and C={2,3}, verify that:  C) = (A × B)  (A × C) how many students recieved medals in exactly (ii) A × ( B 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, and 3 read all the three newspaper. Find (i) the number of people who read at least one of the newspapers, (ii) the number of people who read exactly one newspaper. Q.54 In a class, 18 students offered physics, 23 offered chemistry and 24 offered mathematics. Of these, 13 are in both chemistry and mathematics; 12 in physics and chemistry; 11 in mathematics and physics, and 6 in all the three subjects. Find (i) how many students are there in the class; (ii) how many offered mathematics but not chemistry; (iv) B × B  N : x  3} and B = {x  W, x < 2}, (i) A × (B Q.53 In a survey of 60 people, it was found that 25  3) = (4, 5) Q.57 If A={0,1} and B={1,2,3}, show that A×BB×A. 4 students got exactly two of the three sports, two of the three sports?  2) = (5, 1)  C) = (A × B)  (A × C) Q.62 Let A = {x  W : x < 2}, B = {x  N : 1 < x  4} and C = {3, 5}. Verify that: (i) A × (B  C) = (A × B)  (A × C) (ii) A × (B  C) = (A × B)  (A × C) Q.63 If A×B={(2,3),(-2,4),(0,3),(0,4),(3,3),(3,4)} find A and B. Q.64 Let A = {2, 3} and B = {4, 5}. Find (A × B). How many subsets will (A × B)have ? Q.65 Let A and B be two sets such that n(A) = 3 and n(B) = 2. If a  b  c and (a, 0), (b, 1), (c, 0) are in A × B, find A and B. Q.66 If A = {1, 1}, find (A × A × A). Q.67 If A  B, prove that (A × C)  (B × C). Q.68 Define a relation on a set A. What do you mean by the domain and range of a relation ? Page # 13 SET, RELATIONS AND FUNCTIONS Q.69 If A and B are two sets contain m and n elements Q.78 Let A={2,3,5,7} and B = { 3 , 5 , 9 , 1 3 , 1 5 } . L e t  A, y  B and y=2x1}.Write f in respectively, how many different relations can f={(x,y):x be defined from A to B ? the roster form. Show that f is a function from Q.70Find the domain and range of each of the following relations: (i) R = {(1, 1), (1, 1), (2, 4), (2, 4),(3, 9)}  1   (ii) R =  x,  : x is an int eger, 0  x  6  x     (iii) R = {(x,y):x and y are integers and xy = 4} (iv) R = {(x, y) : x, y  Z and x2 + y2 = 25} Q.71 Let A = {1,2,3, 4, 5, 6}. Define a relation R on A by R = {(x, y) : y = (x + 1)}. (i) Depict R, using arrow diagram. (ii) Write down the domain, co-domain and range of R. Q.72 Let A = {1, 2, 3, 5} and B(4, 6, 9}. Let R = {(x, y) : |x  y| is odd, x  A and y  B}. Write R in the roster form. Q.79 Let f = {(2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)}. Is f a function? Give reason. Q.80 Let g={(1,2),(2,5),(3,8),(4,10),(5,12),(6,12)}. Is g a functions ? If yes, find its domain and range. If no, give reason. Q.81 Let f = {(0, 5), (1, 2), (2, 1), (3, 4), (4, 7)} be a linear function from Z into Z. Find f.  x 2 , when 0  x  2 Q.82 Let f(x) =  and let  2 x, when 2  x  5  x 2 , when 0  x  3 g(x) =  .  2 x, when 3  x  5 Show that f is a function, while g is not a function. Q.73 Let A = {1, 2, 3, 4, 6} and let R = {(a, b) : a, b A to B. Find the domain and range of f.  A and a divides b}. (i) Write R in the roster form. (ii) Find dom (R) and range (R). Q.83 If f(x) = x3, find the value of { f (5)  f (1)} . (5  1)  1 and g(x) = x2 + 1, find: f (i) (f + g) (ii) (f  g) (iii) g Q.84 If f(x) = x Q.74 Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4}. Let R be the relation, ‘is greater than’ from A to B, Write R as a set of ordered pairs.   x2  x , : x  R  be a function from R Q.85 Let f =  2   1  x   Q.75 What is an equivalence relation ? Show that the relation of ‘similarity’ on the set into R. Find range (f). S of all triangles in a plane is an equivalence Q.86 Find the domain of the real-valued function relation. Q.76 Let R = {(a, b) : a, b  N and a = b }. 2 Show that R satisfies none of reflexivity, symmetry and transitivity. Q.77 Let A = {1, 1, 2, 3,) and B = {1, 4, 9, 16} Let f = {(x, y) : x  A, y  B and y = x2}. Show that f is a function from A to B. Find its domain and range. x2  2x  3 f(x) = 2 . x  5x  6 Q.87 Let A = {10, 11, 12, 14, 26}, and Let f : A  N ; f(n) = highest prime factor of n. Find the range of f. Q.88 Find the domain of the real function f(x) = x2  4 . Page # 14 SET, RELATIONS AND FUNCTIONS ANSWER AlCl3(andhydrous) 1. (iii) 2. (i)A= and B(ii)n(A)=6(iii)0W but 0N 3. (i) (ii)  (iii)  (iv)  (v)  (vi)  4. (i) A={L,E,T,R} (ii) B={1,2,3,4,5,6} (iii) C={2,1,0,1,2,3,4} 5. (i) A = {0, 1, 2, 3} (ii) B = {2, 1, 0, 1, 2} (iii) C = {2, 4, 6, 8, 10} 6. (i) infinite (ii) infinite (iii) finite 7. A 8. (i) A = B 9. (i) A and B are equivalent (iii) C and D are equivalent 10. (i) A  B (ii) A  B (iii) A  B 50. 14 51. 25, 35 52. 22 53. (i) 52 (ii) 30 54. (i) 35 (ii) 11 (iii) 11 55. (i) 18 (ii) 3 56. (i) a = 2, b = 3 (ii) a = 5, b = –1 59. (i) A×B={(2,5),(2,7),(3,5),(3,7),(5,5),(5, 7)} (ii) B×A={(5,2),(5,3),(5,5),(7,2)(7,3)(7,5)} (iii) A×A={(2,2),(2,3),(2,5),(3,2),(3,3),(3,5), (5,2), (5, 3), (5, 5)} (iv) B × B ={(5, 5) , (5, 7), (7, 5), (7, 7)} 60. A × B = {(1, 0), (1, 1) (2, 0), (2, 1), (3, 0), (3, 1)} B × A = {(0, 1), (0, 2) (0, 3), (1, 1) (1, 2) (1, 3)}; no 11. (i) P(A)={,{3}} (ii) P(B)={,{–2},{5},{–2,5}} 63. A = {2, 0, 3} and B = {3, 4} 12. P(A) = {, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, 64. A × B = {(2, 4), (2, 5), (3, 4), (3, 5); number of {1,2, 3}} n{P(A)} = 8 13. (i) A = ]4, 0[ subsets of (A × B) = 24 = 16 (ii) B = [0, 3[ (iii) C = ]2, 6] 14. (i) A = {x  R : –2 < x < 2} 66. (A×A×A) = {(–1,1,1), (1,–1,1), (1,1,–1), (ii) B = {x : x R, 5  x  9} (–1,–1,1),(–1,1,–1),(1,–1,–1),(1,1,1),(–1,–1,–1)} (iii) C = {x : x R, 8  x  11} 15. (i) False (ii) True (iii) True 16. (i) {a, b, c, d, e, f, g, h} (ii) {a, c, e, g, h, m, n} (iii) {a, b, c, d, e, f, m, n} 17. (i) {1, 2, 3, 4, 5, 6, 7, 8} (ii) {4, 5, 6, 7, 8, 9, 10, 11} (iii) {1, 2, 3, 4, 5, 7, 8, 9, 10, 11} 18. (i){7,9}(ii){11}(iii)11,13} 19.(i) B(ii) C(iii) D 20. (i) {4, 6} (ii) {2, 4, 5, 6} (iii) {5, 6} 21. (i) {2, 12} (ii) {3, 5, 7} (iii) {2, 3, 5, 7, 12} 22. (i) {a, c} (ii) {d} (iii) {a, b, c, d}  1 1 1 1 1 1 1 1 1 1  23. (i) 1, , , , , , ,  (ii)  , ,   2 3 4 5 6 7 8 2 4 6   1 1 1 1  (iii) 1, , ,  (iv)    3 5 7 8  24. (R  Q) = {x : x  R, x is irrational} 25. (i) {2,3,5,7,11}(ii) 26. (i) B (ii) A (iii)  27. (ii) C and D, since C D= 28. (i) {5,6,7,8,9} (ii){1,3,5,7,9}(iii) {2,3,7,8,9} (iv) {2,4,6,8} (v) {5,7,9} (vi) {2,3,5,6,7,8,9} 30. (i) U (ii)  (iii) A (iv)  43. 12 44. 43 45. (i) 38 (ii) 16 (iii) 14 65. A = {a, b, c} and B = {0, 1} 46. (i) 35 (ii) 30 (iii) 54 47. 60 48. (i) 39 (ii) 11 49. (i) 26 (ii) 18 (iii) 68 69. 2m×n 70. (i) dom (R) = {–2, –1, 1, 2, 3}, range (r) = {1, 4, 9} (ii) dom (R) = {1, 2, 3, 4, 5},  1 1 1 1 range (R) = 1, , , ,   2 3 4 5 (iii) dom (R) = {–4,–2,–1,1,2,4} = range(R) (iv)dom(R) = {–5,–4,–3,0,3,4,5}= range(R) 71. (ii) dom (R) = {1, 2, 3, 4, 5}, co-domain = {1, 2, 3, 4, 5, 6}, range (R) = {2, 3, 4, 5, 6} 72. R ={(1, 4),(1, 6),(2, 9),(3, 4),(3, 6)(5, 4),(5, 6)} 73. (i) R ={(1,1),(1,2),(1,3),(1,4),(1,6),(2,2),(2,4), (2,6),(3,3),(3,6),(4,4),(6,6)} (ii) dom(R)={1,2,3,4,6} and range(R)={1,2,3,4,6} 74. R = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4)} 77. Dom (f) = {1, 1, 2, 3} and range (f) = {1, 4, 9} 78. f = {(2, 3), (3, 5), (5, 9), (7, 13)}, dom (f) = {2, 3, 5, 7}, range (f) = {3, 5, 9, 13}. 79. No, since one element, namely2, has more than one images 80. Yes, dom (g)={1,2,3,4,5,6}, range (g)={2,5,8,10,12} 81. f(x) = 3x 5 83. 31 84. (i) (f+g)(x)=(x+x ) (ii) (f–g)(x) = x – x2 – 2 2 85. [0, 1] 86. R–{2,3} 87. range(f)={3, 5, 7, 11, 13} 88. ], 2]  [2, [