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Test: Sets- 1 - Question 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 ?

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Test: Sets- 1 - Question 2

Consider the set S = {1, ω, ω2}, where ω and w^{2} are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

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Test: Sets- 1 - Question 3

Which one of the following in NOT necessarily a property of a Group?

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Test: Sets- 1 - Question 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?

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Test: Sets- 1 - Question 5

For the composition table of a cyclic group shown below

Q.

Which one of the following choices is correct?

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Test: Sets- 1 - Question 7

Let S be a set of nelements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:

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Test: Sets- 1 - Question 8

How many different non-isomorphic Abelian groups of order 4 are there

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Test: Sets- 1 - Question 9

Consider the set S = {a,b,c,d}. consider the following 4 partitions π_{1}, π_{2}, π_{3}, π_{4} on S : π_{1} = π_{2} = π_{3} = π_{4} = Let ρ be the partial order on the set of partitions S' = {π_{1},π_{2},π_{3}, π_{4}} defined as follows : π_{i } ρ π_{j if and only if }π_{i }refines πj . The poset diagram for (S', ρ) is :

Test: Sets- 1 - Question 10

Consider the set of (column) vector defined by

Which of the following is True ?

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Test: Sets- 1 - Question 11

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:

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Test: Sets- 1 - Question 12

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?

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Test: Sets- 1 - Question 13

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:

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Test: Sets- 1 - Question 14

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}.

Q.

The value of Log_{2}Log_{2}N is ______.

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Test: Sets- 1 - Question 15

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.

**Q. Which one of the following is CORRECT?**

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Test: Sets- 1 - Question 16

Let X and Y be finite sets and f: X -> Y be a function. Which one of the following statements is TRUE?

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Test: Sets- 1 - Question 17

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.

**Q. The size of L is __________.**

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Test: Sets- 1 - Question 18

If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1 ∩ V2 is ______.

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Test: Sets- 1 - Question 19

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

Q.

where e is the identity element. The maximum number of elements in such a group is __________.

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Test: Sets- 1 - Question 20

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.

**Q.
Which one of the following is CORRECT?**

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Test: Sets- 1 - Question 21

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?

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Test: Sets- 1 - Question 22

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?

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Test: Sets- 1 - Question 23

Let S = {1,2,3, ....... , m}, m >3. Let X_{1} ...... X_{n} be subset of S each of size 3. Define a function f from S to the set of atural numbers as, f(i) is the number of sets X_{j } that contain the element i. That is

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Test: Sets- 1 - Question 24

Let A, B and C be non-empty sets and let X = (A - B) - C and Y = (A - C) - (B - C). Which one of the following is TRUE?

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Test: Sets- 1 - Question 25

The following is the Hasse diagram of the poset [{a, b, c, d, e}, ≤]

The poset is

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Test: Sets- 1 - Question 26

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

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Test: Sets- 1 - Question 27

Let R and S be any two equivalence relations on a non-empty set A. Which one of the following statements is TRUE?

Test: Sets- 1 - Question 28

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?

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Test: Sets- 1 - Question 29

What is the minimum number of ordered pairs of non-negative 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"

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Test: Sets- 1 - Question 30

Consider the binary relation:

S = {(x, y) | y = x+1 and x, y ∈ {0, 1, 2, ...}}

**Q. The reflexive transitive closure of S is**

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