11.  Consider the following relation on subsets of the set S of integers between 1 and 2014. For two distinct subsets U and V of S we say U < V if the minimum element in the symmetric difference of the two sets is in U. Consider the following two statements:
S1: There is a subset of S that is larger than every other subset. S2: There is a subset of S that is smaller than every other subset. Which one of the following is CORRECT? 
a.  Both S1 and S2 are true 
b.  S1 is true and S2 is false 
c.  S2 is true and S1 is false 
d.  Neither S1 nor S2 is true 
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Answer: (a).Both S1 and S2 are true

12.  Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L != G and that the size of L is at least 4. The size of L is __________. 
a.  3 
b.  5 
c.  7 
d.  9 
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Answer: (b).5

13.  If V1 and V2 are 4dimensional subspaces of a 6dimensional vector space V, then the smallest possible dimension of V1 ∩ V2 is ______. 
a.  1 
b.  2 
c.  3 
d.  4 
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Answer: (b).2

14.  There are two elements x, y in a group (G,∗) such that every element in the group can be written as a product of some number of x's and y's in some order. It is known that
x ∗ x = y ∗ y = x ∗ y ∗ x ∗ y = y ∗ x ∗ y ∗ x = e where e is the identity element. The maximum number of elements in such a group is __________. 
a.  2 
b.  3 
c.  4 
d.  5 
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Answer: (c).4

15.  Consider the set of all functions f: {0,1, … ,2014} → {0,1, … ,2014} such that f(f(i)) = i, for all 0 ≤ i ≤ 2014. Consider the following statements:
P. For each such function it must be the case that for every i, f(i) = i. Q. For each such function it must be the case that for some i, f(i) = i. R. Each such function must be onto. Which one of the following is CORRECT? 
a.  P, Q and R are true 
b.  Only Q and R are true 
c.  Only P and Q are true 
d.  Only R is true 
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Answer: (b).Only Q and R are true

16.  Let E, F and G be finite sets. Let X = (E ∩ F)  (F ∩ G) and Y = (E  (E ∩ G))  (E  F). Which one of the following is true? 
a.  X ⊂ Y 
b.  X ⊃ Y 
c.  X = Y 
d.  X  Y ≠ φ and Y  X ≠ φ 
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Answer: (c).X = Y

17.  Given a set of elements N = {1, 2, ..., n} and two arbitrary subsets A⊆N and B⊆N, how many of the n! permutations π from N to N satisfy min(π(A)) = min(π(B)), where min(S) is the smallest integer in the set of integers S, and π(S) is the set of integers obtained by applying permutation π to each element of S? 
a.  (n  A ∪ B) A B 
b.  (A^2+B^2)n^2 
c.  n! A∩B / A∪B 
d.  A∩B^2nCA∪B 
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Answer: (c).n! A∩B / A∪B

18.  Let A, B and C be nonempty sets and let X = (A  B)  C and Y = (A  C)  (B  C). Which one of the following is TRUE? 
a.  X = Y 
b.  X ⊂ Y 
c.  Y ⊂ X 
d.  none of these 
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Answer: (a).X = Y

19.  The set {1, 2, 4, 7, 8, 11, 13, 14} is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively 
a.  3 and 13 
b.  2 and 11 
c.  4 and 13 
d.  8 and 14 
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Answer: (c).4 and 13

20.  Let R and S be any two equivalence relations on a nonempty set A. Which one of the following statements is TRUE? 
a.  R ∪ S, R ∩ S are both equivalence relations 
b.  R ∪ S is an equivalence relation 
c.  R ∩ S is an equivalence relation 
d.  Neither R ∪ S nor R ∩ S is an equivalence relation 
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Answer: (c).R ∩ S is an equivalence relation
