Set Theory & Algebra

Question 111

Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ________ . Note - This question was Numerical Type.

• 1
• 5
• 7
• 35

Question 112

A relation R is said to be circular if aRb and bRc together imply cRa. Which of the following options is/are correct?

• If a relation S is reflexive and symmetric, then S is an equivalence relation.
• If a relation S is circular and symmetric, then S is an equivalence relation.
• If a relation S is reflexive and circular, then S is an equivalence relation.
• If a relation S is transitive and circular, then S is an equivalence relation.

Question 113

Let S be a set of consisting of 10 elements. The number of tuples of the form (A,B) such that A and B are subsets of S, and A⊆B is ___________ .

• 59049
• 1024
• 50049
• 59000

Question 114

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

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

Question 115

For the workshops organized in college, students can participate more than one workshop saying that all are organized in different timings. 150 students are enrolled in total for one or more than one workshop. Registration details are as follows : 99 students are enrolled for Cyber Security. 56 students are enrolled for Solar Energy, 89 students are enrolled for Artificial Intelligence. 25 students for both Cyber Security and Solar Energy. 65 students for both Cyber Security and Artificial Intelligence and 18 students for both Solar Energy and Artificial Intelligence. How many students are participating in all three workshops ?

• 12
• 16
• 10
• 14

Question 116

Let f(x) = log|x| and g(x) = sin x . If A is the range of f(g(x)) and B is the range of g(f(x)) then A ∩ B is

• [-1, 0]
• [-1, 0)
• [-∞, 0]
• [-∞,1]

Question 117

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)

Question 118

The ratio of boys to girls in a class is 7 to 3. Among the options below, an acceptable value for the total number of students in the class is:

• 21
• 37
• 50
• 73

Question 119

Given below are two statements 1 and 2, and two conclusions I and II

• Statement 1: All bacteria are microorganisms.
• Statement 2: All pathogens are microorganisms.

• Conclusion I: Some pathogens are bacteria.
• Conclusion II: All pathogens are not bacteria.

Based on the above statements and conclusions, which one of the following options is logically CORRECT?

• Only conclusion I is correct
• Only conclusion II is correct
• Either conclusion I or II is correct
• Neither conclusion I nor II is correct

Question 120

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:

• 22
• 66
• 88
• 110