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Table of contents
1.
Introduction
2.
MCQs
3.
Frequently Asked Questions
4.
Conclusion
Last Updated: Mar 27, 2024
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Set Theory and Algebra

Author Ankit Kumar
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Introduction

Set theory and Algebra are important topics concerning exams or interviews. Thus, it is of utmost importance to practice some critical problems related to the topic. This blog will contain some of the necessary multiple-choice questions with their answers and a brief explanation of those answers.

Let us begin with the MCQs.

MCQs

  1. Consider the set A = {1, ω, ω2}, where ω and w2 are cube roots of unity. If * denotes the multiplication operation, the structure (A, *) forms
    A. A ring
    B. A integral domain
    C. A group
    D. A field
    Ans- Option C
    Explanation- A group is a set of elements together with an operation that combines any two of its elements to form a third element also in the set while satisfying four conditions called the group axioms, namely closure, associativity, identity and invertibility. Here, (A, *) is a group with identity as 1.
     
  2. Let A be a group of 45 elements. Then the largest possible size of a sub-group of A other than A itself is ______
    A. 1
    B. 9
    C. 15
    D. 45
    Ans- Option C
    Explanation- Lagrange’s theorem states that “If H is a subgroup of a finite group G, then the order of subgroup H divides the order of group G.” And the order of subgroup must be a factor of order of group.
    Size of group = O(G) = 45
    Let H be the subgroup of G. Thus, O(H) | O(G)
    The possible order of H is a factor of G, which is 1,3,5,9,15,45.
    The size of the largest possible proper sub-group is 15.
     
  3. Consider the binary relation R = {(a, b), (a, c), (c, a), (c, b)} on the set {a, b, c}. 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
    Ans- Option D
    Explanation- In the relation R = {(a, b), (a, c), (c, a), (c, b)}, (a, b) is present, but (b, a) is not present so it is not symmetric. And both (a, c) and (c, a) are present, so it is not antisymmetric.
     
  4. Let S be a set of k elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:
    A. k and 1
    B. k and k
    C. k^2 and 0
    D. k^2 and k
    Ans- Option D
    Explanation- Equivalence property follows - reflexive, transitive, and symmetric.
    Let's take an example set, S = (a,b,c)
    The largest order set is S x S : {(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)} which are 9 or 3^2 = k^2
    The smallest order set is {(a,a),(b,b),(c,c)} which are 3 or k numbers of element.
     
  5. How many different non-isomorphic Abelian groups of order 8 are there
    A. 2
    B. 3
    C. 4
    D. 5
    Ans- Option A
    Explanation- Here order is 8 i.e. 2^3 and so partition of 3 are {1, 1} and {3, 0}. So the number of different abelian groups is 2. 
     
  6. Calculate the possible number of reflexive relations on a set of 5 elements?
    A. 2^10
    B. 2^15
    C. 2^20
    D. 2^25
    Ans- Option C
    Explanation- Number of elements in a set, n = 5
    Total number of reflexive relations in a set = 2^(n^2 - n) = 2^(5^2 - 5) which is equal to 2^20.
     
  7. 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. Associative but not commutative
    C. Both commutative and associative
    D. Neither commutative nor associative
    Ans- Option A
    Explanation- The binary operation ⊕ defined as : x⊕y = x^2+ y^2 is commutative as y⊕x = y^2 + x^2 = x⊕yit is not associative as x⊕(y⊕z) = x^2 + (y^2+ z^2)^2 and (x⊕y)⊕z = (x^2+ y^2)^2 +z^2
     
  8. What is the maximum number of different Boolean functions involving k Boolean variables?
    A. k^2
    B. 2^k
    C. 2^(2^k)
    D. 2^(k^2)
    Ans- Option C
    Explanation- Total number of inputs sequences possible for a k variable Boolean function = 2^k
    Each input sequence only gives two possible value, either TRUE or FALSE, so the total number of boolean functions are 2*2*2*2*2*2*..........*2*2*2*2, which is 2^k times in total.
    So, 2^(2^k).
     
  9. The set {1,3,5,7,8,9} 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. 3 doesn’t have an inverse
    C. 5 doesn’t have an inverse
    D. 9 doesn’t have an inverse
    Ans- Option B
    Explanation- 3 has an inverse, which is 7
    3*7 mod 10 = 1.
     
  10. Which one of the following is NOT a property of the group?
    A. Commutativity
    B. Associativity
    C. Existence of identity 
    D. Existence of inverse for each element 
    Ans-  Option A
    Explanation- Commutativity is not a property of the group

With this, we mark the end of this blog. I hope it gave you good learning. Let us see some of the FAQs related to Set theory and Algebra.

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Frequently Asked Questions

  1. What is a Set?
    A Set is a collection of objects enclosed in curly braces, separated by commas.
  2. What is Set Theory?
    Set theory deals with sets that are described as a group of objects.
  3. What is an equivalence relation?
    An equivalence relation is symmetric, reflexive and transitive.
  4. List some properties of relations.
    There are three properties of relation: Symmetric, Transitive, and Reflexive.
  5. Define symmetric relation.
    A binary relation R over a set S is symmetric if it holds for all x and y in S that x is related to y if and only if y is related to x.

Let us now summarise our article in the next section.

Conclusion

This article extensively discussed important multiple-choice questions based on the topic of Set theory and Algebra. We also answered all of the questions with a brief explanation. I hope you must have gained a better understanding of the related topic.

If you are not familiar with the topic, then you should definitely read this article to get a better understanding of Set theory and Algebra.

We hope this blog has helped you enhance your knowledge regarding Set theory and Algebra. If you like to learn more, check out our articles on Coding Ninjas Studio. Do upvote our blog to help other ninjas grow. Happy Coding!

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