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2. RELATIONS AND FUNCTIONS CLASS XII MATHEMATICS VOLUME - I JEE 76 A = {1, 2, 3}, the minimum number of ordered pairs which when added to R make it an equivalence relation is (1.1) A) 5 B) 6 C) 7 D) 8 9. For real number x and y, define a relation R, xRy if and only if x y − + 2 is an irrational number. Then the relation R is A) reflexive B) symmetric (1.1) C) transitive D) an equivalence relation 10. Let N denote the set of all natural numbers and R a relation on N × N. Which of the following is an equivalence relation? A) (a, b) R (c, d) if ad (b + c) = bc(a + d) (1.1) B) (a, b) R (c, d) if a + d = b + c C) (a, b) R (c, d) if a d = bc D) All the above 11. Total number of equivalence relations defined in the set S = {a, b, c} is (1.1) A) 5 B) 3 ! C) 23 D) 33 12. On the set N of all natural numbers define the relation R by aRb, if and only if the GCD of a and b is 2, then R is A) reflexive but not symmetric B) only symmetric C) reflexive, and transitive (1.1) D) reflexive, symmetric and transitive 13. Let R be the real line. consider the following subsets of the plane R × R. S = {(x, y): y = x + 1 and 0 < x < 2}, T = {(x, y): x – y is an integer}.Which one of the following is true? A) S is an equivalence relation on R but T is not (1.1) B) T is an equivalence relation on R but S is not C) Neither S nor T is an equivalence relation on R D) Both S and T are equivalence relations on R. 14. If R and S are two non-void relations on a set A.Then, which of the following statements is incorrect? (1.1) A) R and S are transitive implies R ∩ S is transitive B) R and S are transitive implies R ∪ S is transitive C) R and S are symmetric implies R ∪ S is symmetric D) R and S are reflexive implies R ∩ S is reflexive 15. Consider the following relations R = {{x, y}/ x, y are real numbers and x = wy for some rational number w}; S = {(m / n, p / q) / m, n, p and q are integer such that n, q ≠ 0 and qm = pn}, then (1.1) A) neither R or S is an equivalence relation. B) S is an equivalence relation but R is not an equivalence relation C) R and S both are equivalence relations. D) R is an equivalence relation but S is not an equivalence relation. 16. Let X = {1, 2, 3, 4, 5}, the number of different ordered pairs (Y, Z) that can be formed such that Y ⊆ X, Z ⊆ X and Y ∩ Z is empty, is (1.1) A) 35 B) 25 C) 53 D) 52 1. If A = {x : x2 – 3x + 2 = 0}, and R is a universal relation on A, then R is (1.1) A) {(1, 1), (2, 2)} B) {(1, 1)} C) {φ} D) {(1, 1), (1, 2), (2, 1) (2, 2)} 2. If R = {(x, y) | x ∈ N, y ∈ N, x + 3y = 12 }then R-1 is (1.1) A) {(1, 9), (2, 6), (3,3)} B) {(3, 1), (2, 4), (3, 6)} C) {(3, 3), (2, 6), (1, 9)} D) {(1, 3), (1, 6), (1, 9)} 3. If R is the relation ‘less than’ from A = {1, 2, 3, 4, 5}to B = {1, 4}, the set of ordered pairs corresponding to R, then the inverse of R is (1.1) A) {(3,1), (3, 2), (3, 3)} B) {(4,1), (4, 2), (4, 3)} C) {(4, 3), (4, 4), (4, 5)} D) {(1,3), (2, 4), (3, 5)} 4. If A= {1, 2, 3}, B = {1, 4, 6, 9}and R is a relation from A to B defined by 'x is greater than y'. The range of R is A) {1, 4, 6, 9} B) {4, 6, 9} C) {1} D) {4, 6} (1.1) 5. If a relation R is defined on the set Z of integers as follows (a, b) ∈ R ⇔ a2 + b2 = 25. Then, Domain (R) = (1.1) A) {3, 4, 5} B) {0, 3, 4, 5} C) {0, ±3, ±4, ±5} D) {0, ±5} 6. The relation R defined on a set A = {0, -1, 1, 2} by xRy if |x2 + y2 | ≤ 2, then which one of the following is true? (1.1) A) R = {(0, 0), (0, -1), (0, 1), (-1, 0), (-1, 1), (1, 2), (1, 0)} B) R-1 = {(0, 0), (0, -1), (0, 1), (-1, 0), (1, 0)} C) Domain of R is {0, -1, 1, 2} D) Range of R is {0, -1, 1} 7. The number of relations on a set containing 3 elements is A) 9 B) 81 C) 512 D) 1024 (1.1) 8. Let A and B be two sets containing four and two elements respectively. Then the number of subsets of set A × B, each having at least three elements is (1.1) A) 219 B) 256 C) 275 D) 510 9. Two set A and B are as under : A = {(a, b) ∈ R ×R : |a - 5| < 1 and |b - 5| < 1} B = {(a, b) ∈ R ×R : 4(a - 6)2 + 9(b - 5)2 ≤ 36} then A) B ⊂ A B) A ⊂ B (1.1) C) A ∩ B = φ D) neither A⊂ B nor B ⊂ A 10. Le tS be the set of all real numbers. Then the relation R = {(a, b) : 1 + ab > 0} on S is (1.1) A) Reflexive and symmetric but not transitive B) Reflexive and transitive but not symmetric C) Symmetric and transitive but not reflexive D) Equivalence relation 11. The relation R in the set of natural number N is defined by xRY⇔ x2 – 4xy + 3y2 = 0, x, y ∈ N then R is (1.1) A) reflexive but not symmetric and not transitive B) symmetric but not reflexive and not transitive C) transitive but not reflexive and not symmetric D) equivalence relation 12. Let R be the real line. Consider the following subsets of the plane R × R (1.1) S = {(x, y) : y = x + 1 and 0 < x < 2} T = {(x, y) : x – y is an integer}

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