Welcome to the Relations MCQs Page

Dive deep into the fascinating world of Relations with our comprehensive set of Multiple-Choice Questions (MCQs). This page is dedicated to exploring the fundamental concepts and intricacies of Relations, a crucial aspect of Discrete Mathematics. In this section, you will encounter a diverse range of MCQs that cover various aspects of Relations, from the basic principles to advanced topics. Each question is thoughtfully crafted to challenge your knowledge and deepen your understanding of this critical subcategory within Discrete Mathematics.

Check out the MCQs below to embark on an enriching journey through Relations. Test your knowledge, expand your horizons, and solidify your grasp on this vital area of Discrete Mathematics.

Note: Each MCQ comes with multiple answer choices. Select the most appropriate option and test your understanding of Relations. You can click on an option to test your knowledge before viewing the solution for a MCQ. Happy learning!

Relations MCQs | Page 4 of 5

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Q31.

Let a set S = {2, 4, 8, 16, 32} and <= be the partial order defined by S <= R if a divides b. Number of edges in the Hasse diagram of is ______

Discuss

Answer: (b).5

Q32.

The less-than relation, <, on a set of real numbers is ______

not a partial ordering because it is not asymmetric and irreflexive equals antisymmetric

a partial ordering since it is asymmetric and reflexive

a partial ordering since it is antisymmetric and reflexive

not a partial ordering because it is not antisymmetric and reflexive

Discuss

Answer: (a).not a partial ordering because it is not asymmetric and irreflexive equals antisymmetric

Q33.

If the longest chain in a partial order is of length l, then the partial order can be written as _____ disjoint antichains.

l²

l+1

lˡ

Discuss

Answer: (c).l

Q34.

Suppose X = {a, b, c, d} and π1 is the partition of X, π₁ = {{a, b, c}, d}. The number of ordered pairs of the equivalence relations induced by __________

Discuss

Answer: (b).10

Q35.

A partial order P is defined on the set of natural numbers as follows. Here a/b denotes integer division. i)(0, 0) ∊ P. ii)(a, b) ∊ P if and only if a % 10 ≤ b % 10 and (a/10, b/10) ∊ P. Consider the following ordered pairs:

i. (101, 22) ii. (22, 101) iii. (145, 265) iv. (0, 153)

The ordered pairs of natural numbers are contained in P are ______ and ______

(145, 265) and (0, 153)

(22, 101) and (0, 153)

(101, 22) and (145, 265)

(101, 22) and (0, 153)

Discuss

Answer: (d).(101, 22) and (0, 153)

Q36.

The inclusion of ______ sets into R = {{1, 2}, {1, 2, 3}, {1, 3, 5}, {1, 2, 4}, {1, 2, 3, 4, 5}} is necessary and sufficient to make R a complete lattice under the partial order defined by set containment.

{1}, {2, 4}

{1}, {1, 2, 3}

{1}

{1}, {1, 3}, {1, 2, 3, 4}, {1, 2, 3, 5}

Discuss

Answer: (c).{1}

Q37.

Consider the ordering relation a | b ⊆ N x N over natural numbers N such that a | b if there exists c belong to N such that a*c=b. Then ___________

| is an equivalence relation

It is a total order

Every subset of N has an upper bound under |

(N,|) is a lattice but not a complete lattice

Discuss

Answer: (d).(N,|) is a lattice but not a complete lattice

Q38.

Consider the set N* of finite sequences of natural numbers with a denoting that sequence a is a prefix of sequence b. Then, which of the following is true?

Every non-empty subset of has a greatest lower bound

It is uncountable

Every non-empty finite subset of has a least upper bound

Every non-empty subset of has a least upper bound

Discuss

Answer: (a).Every non-empty subset of has a greatest lower bound

Q39.

A partial order ≤ is defined on the set S = {x, b₁, b₂, … bₙ, y} as x ≤ bᵢ for all i and bᵢ ≤ y for all i, where n ≥ 1. The number of total orders on the set S which contain the partial order ≤ is ______

n+4

n²

Discuss

Answer: (c).n!

Q40.

Let (A, ≤) be a partial order with two minimal elements a, b and a maximum element c. Let P:A –> {True, False} be a predicate defined on A. Suppose that P(a) = True, P(b) = False and P(a) ⇒ P(b) for all satisfying a ≤ b, where ⇒ stands for logical implication. Which of the following statements cannot be true?

P(x) = True for all x S such that x ≠ b

P(x) = False for all x ∈ S such that b ≤ x and x ≠ c

P(x) = False for all x ∈ S such that x ≠ a and x ≠ c

P(x) = False for all x ∈ S such that a ≤ x and b ≤ x

Discuss

Answer: (d).P(x) = False for all x ∈ S such that a ≤ x and b ≤ x

Page 4 of 5

Welcome to the Relations MCQs Page

Relations MCQs | Page 4 of 5

Explore more Topics under Discrete Mathematics

Logics and Proofs

Sets and Functions

Sequences,Sums and Matrices

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting Theory

Discrete Probability

Graphs and Graphs Properties

Tree Properties

Boolean Algebra and Modeling Computations

Group Theory

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