Given a boolean function f (x1, x2, ..., xn), which of the following equations is NOT true

f(x1, x2, ..., xn) = f(0, x2, …, xn) + f(1, x2, ..., xn)

f(x1, x2, …, xn) = x1'f(x1, x2, …, xn) + x1f(x1, x2, …, xn)

f(x1, x2, ..., xn) = x2f(x1, x2, …, xn) + x2'f(x1, x2, …, xn)

f(x1, x2, ..., xn) = xn'f(x1, x2, …, 0) + xnf(x1, x2, …,1)

A binary operation &oplus; on a set of integers is defined as x &oplus; y = x2 + y2. Which one of the following statements is TRUE about &oplus;?

Commutative but not associative

Both commutative and associative

Associative but not commutative

Neither commutative nor associative

Set Theory & Algebra

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Question 1

The time complexity of computing the transitive closure of a binary relation on a set of n elements is known to be

O(n*log(n))
O(n^3/2)
O(n³)
O(n)

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Question 2

Given a boolean function f (x₁, x₂, ..., x_n), which of the following equations is NOT true

f(x₁, x₂, …, x_n) = x₁'f(x₁, x₂, …, x_n) + x₁f(x₁, x₂, …, x_n)
f(x₁, x₂, ..., x_n) = x₂f(x₁, x₂, …, x_n) + x₂'f(x₁, x₂, …, x_n)
f(x₁, x₂, ..., x_n) = x_n'f(x₁, x₂, …, 0) + x_nf(x₁, x₂, …,1)
f(x₁, x₂, ..., x_n) = f(0, x₂, …, x_n) + f(1, x₂, ..., x_n)

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Question 3

A binary operation ⊕ on a set of integers is defined as x ⊕ y = x² + y². Which one of the following statements is TRUE about ⊕?

Commutative but not associative
Both commutative and associative
Associative but not commutative
Neither commutative nor associative

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Question 4

Consider the set S = {1, ω, ω2}, where ω and w² are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

A group
A ring
An integral domain
A field

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Question 5

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

Commutativity
Associativity
Existence of inverse for every element
Existence of identity

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Question 6

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?

R is symmetric but NOT antisymmetric
R is NOT symmetric but antisymmetric
R is both symmetric and antisymmetric
R is neither symmetric nor antisymmetric

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Question 7

For the composition table of a cyclic group shown below

Which one of the following choices is correct?

a, b are generators
b, c are generators
c, d are generators
d, a are generators

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Question 8

The number of students in three classes is in the ratio 3:13:6. If 18 students are added to each class, the ratio changes to 15:35:21. The total number of students in all the three classes in the beginning was:

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Question 9

If P, Q, R are subsets of the universal set U, then the value of following expression is

(P ∩ Q ∩ R) ∪ (P^c ∩ Q ∩ R) ∪ Q^c ∪ R^c

Q^c U R^c
P U Q^c U R^c
P^c U Q^c U R^c
U

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Question 10

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

n and n
n² and n
n² and 0
n and 1

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