41.  Let f be a function from a set A to a set B, g a function from B to C, and h a function from A to C, such that h(a) = g(f(a)) for all a ∈ A. Which of the following statements is always true for all such functions f and g? 
a.  g is onto => h is onto 
b.  h is onto => f is onto 
c.  h is onto => g is onto 
d.  h is onto => f and g are onto 
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Answer: (c).h is onto => g is onto

42.  Let A be a set with n elements. Let C be a collection of distinct subsets of A such that for any two subsets S1 and S2 in C, either S1 ⊂ S2 or S2⊂ S1. What is the maximum cardinality of C? 
a.  n 
b.  n + 1 
c.  2n1 + 1 
d.  n! 
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Answer: (b).n + 1

43.  A binary relation R on N x N is defined as follows:
(a, b) R (c, d) if a <= c or b <= d. Consider the following propositions: P: R is reflexive Q: R is transitive Which one of the following statements is TRUE? 
a.  Both P and Q are true 
b.  P is true and Q is false 
c.  P is false and Q is true 
d.  Both P and Q are false 
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Answer: (b).P is true and Q is false

44.  For the set N of natural numbers and a binary operation f : N x N → N, an element z ∊ N is called an identity for f, if f (a, z) = a = f(z, a), for all a ∊ N. Which of the following binary operations have an identity?
1. f (x, y) = x + y  3 2. f (x, y) = max(x, y) 3. f (x, y) = x^y 
a.  I and II only 
b.  II and III only 
c.  I and III only 
d.  None of these 
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Answer: (a).I and II only

45.  Given a boolean function f (x1, x2, ..., xn), which of the following equations is NOT true 
a.  f (x1, x2, ..., xn) = x1'f(x1, x2, ..., xn) + x1f(x1, x2, ..., xn) 
b.  f (x1, x2, ..., xn) = x2f(x1, x2, …, xn) + x2'f(x1, x2, …,xn) 
c.  f (x1, x2, ..., xn) = xn'f(x1, x2, …, 0) + xnf(x1, x2, …,1) 
d.  f (x1, x2, ..., xn) = f(0, x2, …, xn) + f(1, x2, .., xn) 
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Answer: (d).f (x1, x2, ..., xn) = f(0, x2, …, xn) + f(1, x2, .., xn)

46.  Consider the following first order logic formula in which R is a binary relation symbol. ∀x∀y (R(x, y) => R(y, x)) The formula is 
a.  satisfiable and valid 
b.  satisfiable and so is its negation 
c.  unsatisfiable but its negation is valid 
d.  satisfiable but its negation is unsatisfiable 
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Answer: (b).satisfiable and so is its negation

47.  Let P, Q and R be sets let Δ denote the symmetric difference operator defined as PΔQ = (P U Q)  (P ∩ Q). Using Venn diagrams, determine which of the following is/are TRUE? PΔ (Q ∩ R) = (P Δ Q) ∩ (P Δ R) P ∩ (Q ∩ R) = (P ∩ Q) Δ (P Δ R) 
a.  I only 
b.  II only 
c.  Neither I nor II 
d.  Both I and II 
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Answer: (c).Neither I nor II

48.  What is the cardinality of the set of integers X defined below? X = {n  1 ≤ n ≤ 123, n is not divisible by either 2, 3 or 5} ? 
a.  28 
b.  33 
c.  37 
d.  44 
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Answer: (b).33

49.  Let A = {a, b, c, d }, B = { p, q, r, s } denote sets. R : A –> B, R is a function from A to B. Then which of the following relations are not functions ?
(i) { (a, p) (b, q) (c, r) } (ii) { (a, p) (b, q) (c, s) (d, r) } (iii) { (a, p) (b, s) (b, r) (c, q) } 
a.  (i) and (ii) only 
b.  (ii) and (iii) only 
c.  (i) and (iii) only 
d.  None of these 
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Answer: (c).(i) and (iii) only

50.  Let A = { 1,2,3,4,…….∞ } and a binary operation ‘+’ is defined by a + b = ab ∀ a,b ∈ A. Which of the following is true ? 
a.  ( A, + ) is a semi group but not monoid 
b.  ( A, + ) is a monoid but not group 
c.  ( A, + ) is a group 
d.  ( A, + ) is not a semi group 
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Answer: (b).( A, + ) is a monoid but not group
