Welcome to the Logics and Proofs MCQs Page

Dive deep into the fascinating world of Logics and Proofs with our comprehensive set of Multiple-Choice Questions (MCQs). This page is dedicated to exploring the fundamental concepts and intricacies of Logics and Proofs, a crucial aspect of Discrete Mathematics. In this section, you will encounter a diverse range of MCQs that cover various aspects of Logics and Proofs, 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 Logics and Proofs. 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 Logics and Proofs. You can click on an option to test your knowledge before viewing the solution for a MCQ. Happy learning!

Logics and Proofs MCQs | Page 10 of 12

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Q91.

Translate ∀x∃y(x < y) in English, considering domain as a real number for both the variable.

For all real number x there exists a real number y such that x is less than y

For every real number y there exists a real number x such that x is less than y

For some real number x there exists a real number y such that x is less than y

For each and every real number x and y such that x is less than y

Discuss

Answer: (a).For all real number x there exists a real number y such that x is less than y

Q92.

“The product of two negative real numbers is not negative.” Is given by?

∃x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))

∃x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))

∀x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))

∀x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))

Discuss

Answer: (d).∀x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))

Q93.

Let Q(x, y) be the statement “x + y = x − y.” If the domain for both variables consists of all integers, what is the truth value of ∃xQ(x, 4).

True

False

May be True or False

Can't say

Discuss

Answer: (b).False

Q94.

Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use quantifiers to express, “Joy is loved by everyone.”

∀x L(x, Joy)

∀y L(Joy,y)

∃y∀x L(x, y)

∃x ¬L(Joy, x)

Discuss

Answer: (a).∀x L(x, Joy)

Q95.

Let T (x, y) mean that student x likes dish y, where the domain for x consists of all students at your school and the domain for y consists of all dishes. Express ¬T (Amit, South Indian) by a simple English sentence.

All students does not like South Indian dishes.

Amit does not like South Indian people.

Amit does not like South Indian dishes.

Amit does not like some dishes.

Discuss

Answer: (d).Amit does not like some dishes.

Q96.

Express, “The difference of a real number and itself is zero” using required operators.

∀x(x − x! = 0)

∀x(x − x = 0)

∀x∀y(x − y = 0)

∃x(x − x = 0)

Discuss

Answer: (b).∀x(x − x = 0)

Q97.

Use quantifiers and predicates with more than one variable to express, “There is a pupil in this lecture who has taken at least one course in Discrete Maths.”

∃x∃yP (x, y), where P (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all Discrete Maths lectures

∃x∃yP (x, y), where P (x, y) is “x has taken y,” the domain for x consists of all Discrete Maths lectures, and the domain for y consists of all pupil in this class

∀x∀yP(x, y), where P (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all Discrete Maths lectures

∃x∀yP(x, y), where P (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all Discrete Maths lectures

Discuss

Answer: (a).∃x∃yP (x, y), where P (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all Discrete Maths lectures

Q98.

Determine the truth value of ∃n∃m(n + m = 5 ∧ n − m = 2) if the domain for all variables consists of all integers.

True

False

May be True or False

Can't say

Discuss

Answer: (b).False

Q99.

Find a counter example of ∀x∀y(xy > y), where the domain for all variables consists of all integers.

x = -1, y = 17

x = -2 y = 8

Both x = -1, y = 17 and x = -2 y = 8

Does not have any counter example

Discuss

Answer: (c).Both x = -1, y = 17 and x = -2 y = 8

Q100.

Which rule of inference is used in each of these arguments, “If it is Wednesday, then the Smartmart will be crowded. It is Wednesday. Thus, the Smartmart is crowded.”

Modus tollens

Modus ponens

Disjunctive syllogism

Simplification

Discuss

Answer: (b).Modus ponens

Page 10 of 12

Welcome to the Logics and Proofs MCQs Page

Logics and Proofs MCQs | Page 10 of 12

Explore more Topics under Discrete Mathematics

Sets and Functions

Sequences,Sums and Matrices

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting Theory

Discrete Probability

Relations

Graphs and Graphs Properties

Tree Properties

Boolean Algebra and Modeling Computations

Group Theory

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