Test: Set Theory & Algebra- 2

QUESTION: 1

Some group (G, o) is known to be abelian. Then, which one of the following is true for G?

A.
B.
C.
D.
G is of finite order

Solution:

Since group is abelian so it is commutative as well as associative.

QUESTION: 2

Let R be a sym m etric and transitive relation on a set A Then

A.
R is reflexive and hence an equivalence relation
B.
R is reflexive and hence a partial order
C.
R is reflexive and hence not an equivalence relation
D.
None of the above

Solution:

A relation which is symmetric and transitive, need not foe reflexive relation.
(i) R = {}; on the set A = {a, b}. The relation R is symmetric and transitive but not reflexive.
(ii) R = {(a, a), (b, b)}; on the set A = {a, b}.
The relation R is symmetric, transitive and also reflexive.
∴ A relation is transitive and symmetric relation but need not be reflexive relation.

QUESTION: 3

Let A and B be sets and let A^c and B^c denote the complements of the sets A and B. The set (A - B) ∪ (B - A) ∪ (A ∩ B) is equal to

A.
A ∪ B
B.
A^c ∪ B^c
C.
A ∩ B
D.
A^c ∩ B^c

Solution:

Representing above set using Venn diagram as follows:

∴ By Venn diagram,

QUESTION: 4

Which one of the following is false?

A.
The set of all bijective functions on a finite setforms a group under function composition.
B.
The set { 1 , 2 ,. . . , p - 1} forms a group under multiplication mod p where p is a prime number.
C.
The set of all strings over a finite alphabet ∑ forms a group under concatenation.
D.
A subset of G is a subgroup of the group (G, *) if and only if for any pair of elements a, b ∈ s, a * b^-1 ∈ s.

Solution:

In option (c), the set of all strings over a finite alphabet ∑ doesn’t forms a group under concatenation because the inverse of a string doesn’t exist with respect to concatenation.

QUESTION: 5

The binary relation R = {(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)},on the set A = {1 , 2 , 3, 4} is

A.
Reflexive, symmetric and transitive
B.
Neither reflexive, nor irreflexive but transitive
C.
Irreflexive, symmetric and transitive
D.
Irreflexive and antisymmetric

Solution:

The relation R doesn’t contain (4, 4), so R is not reflexive relation.
Since relation R contains ( 1 ,1 ) , (2, 2) and (3, 3).
Therefore, relation R is also not irreflexive.
That R is transitive, can be checked by systematically checking for all (a, b) and (b, c) in R, whether (a, c) also exists in R.

QUESTION: 6

Let' P(S) denote the power set of a set S. Which of the following is always true?

A.
P(P(S)) = P(S)
B.
P(S) ∩ P(P(S)) = {φ}
C.
P(S) ∩ S = P(S)
D.
S ∉ P(S)

Solution:

φ always present in any powerset of a set and φ is the only common element between P(S) and P(P(S)

QUESTION: 7

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 non-commutative group
C.
Does not have a right identity element
D.
Forms a group if the empty string is removed from ∑*

Solution:

QUESTION: 8

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, ...}}

Solution:

QUESTION: 9

In a class of 200 students, 125 students have taken Programming Language course, 85 students have taken Data Structures course, 65 students have taken Computer Organization course; 50 students have taken both Programming Language and Data Structures, 35 students have taken both Data Structures and Computer Organization; 30 students have taken both Data Structures and Computer Organization, 15 students have taken all the three courses. How many students have not taken any of the three courses?

A.
15
B.
20
C.
25
D.
30

Solution:

According to inclusion - exclusion formula:

PL = students who have taken programming.
DS = Students who have taken Data structures.
CO = Students who have taken Computer Organization.
So, the number of students who have taken atleast 1 of the 3 courses is given by:
= 125 + 85 + 65 - 50 - 35 - 30 + 15
= 175
Therefore, the number of students who have not taken any of the 3 courses is
= Total students-students taken atleast 1 course
= 200-175 = 25

QUESTION: 10

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?

A.
X = Y
B.
X ⊂ Y
C.
Y ⊂ X
D.
None of these

Solution:

QUESTION: 11

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

Solution:

QUESTION: 12

A relation R fs defined on ordered pairs of integers as follows: (x, y) R (u, v) if x < u and y > v. 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

Solution:

QUESTION: 13

Let P, Q and R be sets let A denote the symmetric difference operator defined as PΔG = ( P ∪ Q) - (P ∩ Q). Using Venn diagrams, determine which of the following is/are TRUE?

A.
I only
B.
II only
C.
Neither I nor II
D.
Both I and II

Solution:

P Δ Q = PQ' + PQ
where Δ is symmetric difference between P and Q.
1.

2.

QUESTION: 14

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

Solution:

∴ R is not antisymmetric.
R is neither-symmetric nor anti-symmetric.

QUESTION: 15

How many onto (or surjective) functions are there from an n-element (n ≥ 2) set to a 2-element set?

A.
2ⁿ
B.
2ⁿ - 1
C.
2ⁿ - 2
D.
2 (2ⁿ- 2 )

Solution:

Let n = 2. There are only 2 onto functions as shown below:

The number of onto functions from a set A with m elements to set B with n elements where n < m is given by

15 Questions MCQ Test GATE Electrical Engineering (EE) 2023 Mock Test Series | Test: Set Theory & Algebra- 2

Some group (G, o) is known to be abelian. Then, which one of the following is true for G?

Let R be a sym m etric and transitive relation on a set A Then

Let A and B be sets and let A^c and B^c denote the complements of the sets A and B. The set (A - B) ∪ (B - A) ∪ (A ∩ B) is equal to

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