21.  Let f: B → C and g: A → B be two functions and let h = f o g. Given that h is an onto function. Which one of the following is TRUE? 
a.  f and g should both be onto functions 
b.  f should be onto but g need not be onto 
c.  g should be onto but f need not be onto 
d.  both f and g need not be onto 
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Answer: (b).f should be onto but g need not be onto

22.  What is the minimum number of ordered pairs of nonnegative numbers that should be chosen to ensure that there are two pairs (a, b) and (c, d) in the chosen set such that "a ≡ c mod 3" and "b ≡ d mod 5" . 
a.  4 
b.  6 
c.  16 
d.  24 
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Answer: (c).16

23.  Consider the binary relation:
S = {(x, y)  y = x+1 and x, y ∈ {0, 1, 2, ...}} The reflexive transitive closure of S is 
a.  {(x, y)  y > x and x, y ∈ {0, 1, 2, ... }} 
b.  {(x, y)  y ≥ x and x, y ∈ {0, 1, 2, ... }} 
c.  {(x, y)  y < x and x, y ∈ {0, 1, 2, ... }} 
d.  {(x, y)  y ≤ x and x, y ∈ {0, 1, 2, ... }} 
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Answer: (b).{(x, y)  y ≥ x and x, y ∈ {0, 1, 2, ... }}

24.  The following is the incomplete operation table a 4element group.
* e a b c e e a b c a a b c e b c The last row of the table is 
a.  c a e b 
b.  c b a e 
c.  c b e a 
d.  c e a b 
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Answer: (d).c e a b

25.  The inclusion of which of the following sets into
S = {{1, 2}, {1, 2, 3}, {1, 3, 5}, (1, 2, 4), (1, 2, 3, 4, 5}} is necessary and sufficient to make S a complete lattice under the partial order defined by set containment ? 
a.  {1} 
b.  {1}, {2, 3} 
c.  {1}, {1, 3} 
d.  {1}, {1, 3}, (1, 2, 3, 4}, {1, 2, 3, 5) 
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Answer: (a).{1}

26.  Consider the set ∑* of all strings over the alphabet ∑ = {0, 1}. ∑* with the concatenation operator for strings 
a.  does not form a group 
b.  forms a noncommutative group 
c.  does not have a right identity element 
d.  forms a group if the empty string is removed from ∑* 
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Answer: (a).does not form a group

27.  Let (5, ≤) be a partial order with two minimal elements a and b, and a maximum element c.
Let P : S → {True, False} be a predicate defined on S. Suppose that P(a) = True, P(b) = False and P(x) ⇒ P(y) for all x, y ∈ S satisfying x ≤ y, where ⇒ stands for logical implication. Which of the following statements CANNOT be true ? 
a.  P(x) = True for all x ∈ S such that x ≠ b 
b.  P(x) = False for all x ∈ S such that x ≠ a and x ≠ c 
c.  P(x) = False for all x ∈ S such that b ≤ x and x ≠ c 
d.  P(x) = False for all x ∈ S such that a ≤ x and b ≤ x 
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Answer: (d).P(x) = False for all x ∈ S such that a ≤ x and b ≤ x

28.  Let f : A → B be an injective (onetoone) function.
Define g : 2^A → 2^B as : g(C) = {f(x)  x ∈ C}, for all subsets C of A. Define h : 2^B → 2^A as : h(D) = {x  x ∈ A, f(x) ∈ D}, for all subsets D of B. Which of the following statements is always true ? 
a.  g(h(D)) ⊆ D 
b.  g(h(D)) ⊇ D 
c.  g(h(D)) ∩ D = ф 
d.  g(h(D)) ∩ (B  D) ≠ ф 
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Answer: (a).g(h(D)) ⊆ D

29.  Which of the following is true? 
a.  The set of all rational negative numbers forms a group under multiplication 
b.  The set of all nonsingular matrices forms a group under multiplication 
c.  The set of all matrices forms a group under multiplication 
d.  Both (2) and (3) are true 
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Answer: (b).The set of all nonsingular matrices forms a group under multiplication

30.  The binary relation S = ф (empty set) on set A = {1, 2, 3} is : 
a.  Neither reflexive nor symmetric 
b.  Symmetric and reflexive 
c.  Transitive and reflexive 
d.  Transitive and symmetric 
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Answer: (d).Transitive and symmetric
