1.  A binary operation on a set of integers is defined as x y = x^2 + y^2. Which one of the following statements is TRUE about ? 
a.  Commutative but not associative 
b.  Both commutative and associative 
c.  Associative but not commutative 
d.  Neither commutative nor associative 
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Answer: (a).Commutative but not associative

2.  Consider the set S = {1, ω, ω^2}, where ω and w^2 are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms 
a.  A group 
b.  A ring 
c.  An integral domain 
d.  A field 
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Answer: (a).A group

3.  Which one of the following in NOT necessarily a property of a Group? 
a.  Commutativity 
b.  Associativity 
c.  Existence of inverse for every element 
d.  Existence of identity 
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Answer: (a).Commutativity

4.  Consider the binary relation R = {(x, y), (x, z), (z, x), (z, y)} on the set {x, y, z}. Which one of the following is TRUE? 
a.  R is symmetric but NOT antisymmetric 
b.  R is NOT symmetric but antisymmetric 
c.  R is both symmetric and antisymmetric 
d.  R is neither symmetric nor antisymmetric 
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Answer: (d).R is neither symmetric nor antisymmetric

5.  Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are: 
a.  n and n 
b.  n^2 and n 
c.  n^2 and 0 
d.  n and 1 
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Answer: (b).n^2 and n

6.  How many different nonisomorphic Abelian groups of order 4 are there 
a.  2 
b.  3 
c.  4 
d.  5 
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Answer: (a).2

7.  Let X, Y, Z be sets of sizes x, y and z respectively. Let W = X x Y. Let E be the set of all subsets of W. The number of functions from Z to E is: 
a.  z^2^xy 
b.  z x 2^xy 
c.  z^2^(x + y) 
d.  2^xyz 
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Answer: (d).2^xyz

8.  The set {1, 2, 3, 5, 7, 8, 9} under multiplication modulo 10 is not a group. Given below are four plausible reasons. Which one of them is false? 
a.  It is not closed 
b.  2 does not have an inverse 
c.  3 does not have an inverse 
d.  8 does not have an inverse 
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Answer: (c).3 does not have an inverse

9.  A relation R is defined on ordered pairs of integers as follows: (x,y) R(u,v) if x < u and y > v. Then R is: Then R is: 
a.  Neither a Partial Order nor an Equivalence Relation 
b.  A Partial Order but not a Total Order 
c.  A Total Order 
d.  An Equivalence Relation 
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Answer: (a).Neither a Partial Order nor an Equivalence Relation

10.  Let S denote the set of all functions f: {0,1}^4 > {0,1}. Denote by N the number of functions from S to the set {0,1}. The value of Log2Log2N is ______. 
a.  12 
b.  13 
c.  15 
d.  16 
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Answer: (d).16
