31.  Consider the following relations:
R1(a,b) iff (a+b) is even over the set of integers R2(a,b) iff (a+b) is odd over the set of integers R3(a,b) iff a.b > 0 over the set of nonzero rational numbers R4(a,b) iff a  b <= 2 over the set of natural numbers Which of the following statements is correct? 
a.  R1 and R2 are equivalence relations, R3 and R4 are not 
b.  R1 and R3 are equivalence relations, R2 and R4 are not 
c.  R1 and R4 are equivalence relations, R2 and R3 are not 
d.  R1, R2, R3 and R4 are all equivalence relations 
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Answer: (b).R1 and R3 are equivalence relations, R2 and R4 are not

32.  Consider the following statements:
S1: There exists infinite sets A, B, C such that A ∩ (B ∪ C) is finite. S2: There exists two irrational numbers x and y such that (x+y) is rational. Which of the following is true about S1 and S2? 
a.  Only S1 is correct 
b.  Only S2 is correct 
c.  Both S1 and S2 are correct 
d.  None of S1 and S2 is correct 
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Answer: (c).Both S1 and S2 are correct

33.  A relation R is defined on the set of integers as xRy if f(x + y) is even. Which of the following statement is true? 
a.  R is not an equivalence relation 
b.  R is an equivalence relation having 1 equivalence class 
c.  R is an equivalence relation having 2 equivalence classes 
d.  R is an equivalence relation having 3 equivalence classes 
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Answer: (c).R is an equivalence relation having 2 equivalence classes

34.  Let R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is True? 
a.  R is symmetric and reflexive but not transitive 
b.  R is reflexive but not symmetric and not transitive 
c.  R is transitive but not reflexive and not symmetric 
d.  R is symmetric but not reflexive and not transitive 
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Answer: (d).R is symmetric but not reflexive and not transitive

35.  The cardinality of the power set of {0, 1, 2 . . ., 10} is _________. 
a.  1024 
b.  1023 
c.  2048 
d.  2043 
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Answer: (c).2048

36.  Consider two relations R1(A, B) with the tuples (1, 5), (3, 7) and R1(A, C) = (1, 7), (4, 9). Assume that R(A,B,C) is the full natural outer join of R1 and R2. Consider the following tuples of the form (A,B,C)
a = (1, 5, null), b = (1, null, 7), c = (3, null, 9), d = (4, 7, null), e = (1, 5, 7), f = (3, 7, null), g = (4, null, 9). Which one of the following statements is correct? 
a.  R contains a, b, e, f, g but not c, d 
b.  R contains a, b, c, d, e, f, g 
c.  R contains e, f, g but not a, b 
d.  R contains e but not f, g 
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Answer: (c).R contains e, f, g but not a, b

37.  The number of onto functions (surjective functions) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is ________________ 
a.  36 
b.  64 
c.  81 
d.  72 
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Answer: (a).36

38.  Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X and Y. Let f be randomly chosen from F. The probability of f being onetoone is _________. 
a.  0.95 
b.  0.80 
c.  0.75 
d.  0.70 
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Answer: (a).0.95

39.  Let R be a relation on the set of ordered pairs of positive integers such that ((p, q), (r, s)) ∈ R if and only if p–s = q–r. Which one of the following is true about R? 
a.  Both reflexive and symmetric 
b.  Reflexive but not symmetric 
c.  Not reflexive but symmetric 
d.  Neither reflexive nor symmetric 
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Answer: (c).Not reflexive but symmetric

40.  Let R1 be a relation from A = {1, 3, 5, 7} to B = {2, 4, 6, 8} and R2 be another relation from B to C = {1, 2, 3, 4} as defined below:
1. An element x in A is related to an element y in B (under R1) if x + y is divisible by 3. 2. An element x in B is related to an element y in C (under R2) if x + y is even but not divisible by 3. Which is the composite relation R1R2 from A to C? 
a.  R1R2 = {(1, 2), (1, 4), (3, 3), (5, 4), (7, 3)} 
b.  R1R2 = {(1, 2), (1, 3), (3, 2), (5, 2), (7, 3)} 
c.  R1R2 = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)} 
d.  R1R2 = {(3, 2), (3, 4), (5, 1), (5, 3), (7, 1)} 
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Answer: (c).R1R2 = {(1, 2), (3, 2), (3, 4), (5, 4), (7, 2)}
