“Parul is out for a trip or it is not snowing” and “It is snowing or Raju is playing chess” imply that __________

Parul is out for a trip and Raju is playing chess

Parul is out for a trip or Raju is playing chess

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.

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 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 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

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

Does not have any counter example

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

Does not have any counter example

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

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

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

What is the contrapositive of the conditional statement? “The home team misses whenever it is drizzling?”

If it is drizzling, then home team misses

If the home team misses, then it is drizzling

If it is not drizzling, then the home team does not misses

If the home team wins, then it is not drizzling

What is the converse of the conditional statement “If it ices today, I will play ice hockey tomorrow.”

“If I do not play ice hockey tomorrow, then it will not have iced today.”

“I will play ice hockey tomorrow only if it ices today.”

“If I do not play ice hockey tomorrow, then it will not have iced today.”

“If it does not ice today, then I will not play ice hockey tomorrow.”

“I will not play ice hockey tomorrow only if it ices today.”

What are the converse of the conditional statement “When Raj stay up late, it is necessary that Raj sleep until noon.”

“If Raj stay up late, then Raj sleep until noon.”

“If Raj does not stay up late, then Raj does not sleep until noon.”

“If Raj does not sleep until noon, then Raj does not stay up late.”

“If Raj sleep until noon, then Raj stay up late.”

What are the contrapositive of the conditional statement “I come to class whenever there is going to be a test.”

“If there is not going to be a test, then I don’t come to class.”

“If I come to class, then there will be a test.”

“If I do not come to class, then there will not be a test.”

“If there is not going to be a test, then I don’t come to class.”

“If there is going to be a test, then I don’t come to class.”

What are the inverse of the conditional statement “ A positive integer is a composite only if it has divisors other than 1 and itself.”

“A positive integer is a composite if it has divisors other than 1 and itself.”

“If a positive integer has no divisors other than 1 and itself, then it is not composite.”

“If a positive integer is not composite, then it has no divisors other than 1 and itself.”

What are the contrapositive of the conditional statement “Medha will find a decent job when she labour hard.”?

“If Medha will find a decent job, then she labour hard.”

“If Medha labour hard, then she will find a decent job.”

“If Medha will not find a decent job, then she not labour hard.”

“If Medha will find a decent job, then she labour hard.”

“If Medha not labour hard, then she will not find a decent job.”

What are the inverse of the conditional statement “If you make your notes, it will be a convenient in exams.”

“If it will not be a convenient in exams, then you did not make your notes.”

“If you make notes, then it will be a convenient in exams.”

“If you do not make notes, then it will not be a convenient in exams.”

“If it will not be a convenient in exams, then you did not make your notes.”

“If it will be a convenient in exams, then you make your notes

Which of the arguments is not valid in proving sum of two odd number is not odd.

2n +1 + 2m +1 = 2(n+m+1) hence true for all

Let domain of m includes all students, P (m) be the statement “m spends more than 2 hours in playing polo”. Express ∀m ¬P (m) quantification in English.

A student is there who spends more than 2 hours in playing polo

There is a student who does not spend more than 2 hours in playing polo

All students spends more than 2 hours in playing polo

No student spends more than 2 hours in playing polo

”Everyone wants to learn cosmology.” This argument may be true for which domains?

All students in your cosmology class

All the cosmology learning students in the world

107 + Mcqs in Logics and Proofs in Discrete Mathematics Page 1 McqOptions

1.	What rule of inference is used in this argument?
A.	Modus tollens
B.	Modus ponens
C.	Disjunctive syllogism
D.	Hypothetical syllogism
Answer» E.

2.	“Parul is out for a trip or it is not snowing” and “It is snowing or Raju is playing chess” imply that __________
A.	Parul is out for trip
B.	Raju is playing chess
C.	Parul is out for a trip and Raju is playing chess
D.	Parul is out for a trip or Raju is playing chess
Answer» E.

3.	Let the statement be “If n is not an odd integer then square of n is not odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “(square of n) is not odd.” For direct proof we should prove _________
A.	∀nP ((n) → Q(n))
B.	∃ nP ((n) → Q(n))
C.	∀n~(P ((n)) → Q(n))
D.	∀nP ((n) → ~(Q(n)))
Answer» B. ∃ nP ((n) → Q(n))

4.	Which rule of inference is used, ”Bhavika will work in an enterprise this summer. Therefore, this summer Bhavika will work in an enterprise or he will go to beach.”
A.	Simplification
B.	Conjunction
C.	Addition
D.	Disjunctive syllogism
Answer» D. Disjunctive syllogism

5.	The premises (p ∧ q) ∨ r and r → s imply which of the conclusion?
A.	p ∨ r
B.	p ∨ s
C.	p ∨ q
D.	q ∨ r
Answer» C. p ∨ q

6.	Which rule of inference is used in each of these arguments, “If it hailstoday, the local office will be closed. The local office is not closed today. Thus, it did not hailed today.”
A.	Modus tollens
B.	Conjunction
C.	Hypothetical syllogism
D.	Simplification
Answer» B. Conjunction

8.	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.”
A.	∀x L(x, Joy)
B.	∀y L(Joy,y)
C.	∃y∀x L(x, y)
D.	∃x ¬L(Joy, x)
Answer» B. ∀y L(Joy,y)

9.	Express, “The difference of a real number and itself is zero” using required operators.
A.	∀x(x − x! = 0)
B.	∀x(x − x = 0)
C.	∀x∀y(x − y = 0)
D.	∃x(x − x = 0)
Answer» C. ∀x∀y(x − y = 0)

11.	Find a counter example of ∀x∀y(xy > y), where the domain for all variables consists of all integers.
A.	x = -1, y = 17
B.	x = -2 y = 8
C.	Both x = -1, y = 17 and x = -2 y = 8
D.	Does not have any counter example
Answer» D. Does not have any counter example

12.	Determine the truth value of ∃n∃m(n + m = 5 ∧ n − m = 2) if the domain for all variables consists of all integers.
A.	True
B.	False
C.	May be True or False
D.	Can't say
Answer» C. May be True or False

15.	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).
A.	True
B.	False
C.	May be True or False
D.	Can't say
Answer» C. May be True or False

16.	The inverse of p → q is the proposition of ____________
A.	¬p → ¬q
B.	¬q → ¬p
C.	q → p
D.	¬q → p
Answer» B. ¬q → ¬p

17.	“The product of two negative real numbers is not negative.” Is given by?
A.	∃x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
B.	∃x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
C.	∀x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
D.	∀x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
Answer» E.

27.	A proof broken into distinct cases, where these cases cover all prospects, such proofs are known as ___________
A.	Direct proof
B.	Contrapositive proofs
C.	Vacuous proof
D.	Proof by cases
Answer» D. Proof by cases

29.	A proof that p → q is true based on the fact that q is true, such proofs are known as ___________
A.	Direct proof
B.	Contrapositive proofs
C.	Trivial proof
D.	Proof by cases
Answer» D. Proof by cases

18.	The contrapositive of p → q is the proposition of ____________
A.	¬p → ¬q
B.	¬q → ¬p
C.	q → p
D.	¬q → p
Answer» C. q → p

20.	The converse of p → q is the proposition of _______________
A.	¬p → ¬q
B.	¬q → ¬p
C.	q → p
D.	¬q → p
Answer» D. ¬q → p

28.	Which of the arguments is not valid in proving sum of two odd number is not odd.
A.	3 + 3 = 6, hence true for all
B.	2n +1 + 2m +1 = 2(n+m+1) hence true for all
C.	All of the mentioned
D.	None of the mentioned
Answer» B. 2n +1 + 2m +1 = 2(n+m+1) hence true for all

30.	A theorem used to prove other theorems is known as _______________
A.	Lemma
B.	Corollary
C.	Conjecture
D.	None of the mentioned
Answer» B. Corollary

31.	Which of the following can only be used in disproving the statements?
A.	Direct proof
B.	Contrapositive proofs
C.	Counter Example
D.	Mathematical Induction
Answer» D. Mathematical Induction

33.	When to proof P→Q true, we proof P false, that type of proof is known as ___________
A.	Direct proof
B.	Contrapositive proofs
C.	Vacuous proof
D.	Mathematical Induction
Answer» D. Mathematical Induction

34.	A proof covering all the possible cases, such type of proofs are known as ___________
A.	Direct proof
B.	Proof by Contradiction
C.	Vacuous proof
D.	Exhaustive proof
Answer» E.

35.	A compound proposition that is always ___________ is called a tautology.
A.	True
B.	False
C.	May be True or False
D.	Can't say
Answer» B. False

36.	In proving √5 as irrational, we begin with assumption √5 is rational in which type of proof?
A.	Direct proof
B.	Proof by Contradiction
C.	Vacuous proof
D.	Mathematical Induction
Answer» C. Vacuous proof

38.	(A ∨ ¬A) ∨ (q ∨ T) is a __________
A.	Tautology
B.	Contradiction
C.	Contingency
D.	None of the mentioned
Answer» B. Contradiction

Explore topic-wise MCQs in Discrete Mathematics.

What rule of inference is used in this argument?

“Parul is out for a trip or it is not snowing” and “It is snowing or Raju is playing chess” imply that __________

Let the statement be “If n is not an odd integer then square of n is not odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “(square of n) is not odd.” For direct proof we should prove _________

Which rule of inference is used, ”Bhavika will work in an enterprise this summer. Therefore, this summer Bhavika will work in an enterprise or he will go to beach.”

The premises (p ∧ q) ∨ r and r → s imply which of the conclusion?

Which rule of inference is used in each of these arguments, “If it hailstoday, the local office will be closed. The local office is not closed today. Thus, it did not hailed today.”

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.

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.”

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

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.”

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

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

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.”

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

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).

The inverse of p → q is the proposition of ____________

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

The contrapositive of p → q is the proposition of ____________

What is the contrapositive of the conditional statement? “The home team misses whenever it is drizzling?”

The converse of p → q is the proposition of _______________

What is the converse of the conditional statement “If it ices today, I will play ice hockey tomorrow.”

What are the converse of the conditional statement “When Raj stay up late, it is necessary that Raj sleep until noon.”

What are the contrapositive of the conditional statement “I come to class whenever there is going to be a test.”

What are the inverse of the conditional statement “ A positive integer is a composite only if it has divisors other than 1 and itself.”

What are the contrapositive of the conditional statement “Medha will find a decent job when she labour hard.”?

What are the inverse of the conditional statement “If you make your notes, it will be a convenient in exams.”

A proof broken into distinct cases, where these cases cover all prospects, such proofs are known as ___________

Which of the arguments is not valid in proving sum of two odd number is not odd.

A proof that p → q is true based on the fact that q is true, such proofs are known as ___________

A theorem used to prove other theorems is known as _______________

Which of the following can only be used in disproving the statements?

Let the statement be “If n is not an odd integer then sum of n with some not odd number will not be odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “sum of n with some not odd number will not be odd.” A proof by contraposition will be ________

When to proof P→Q true, we proof P false, that type of proof is known as ___________

A proof covering all the possible cases, such type of proofs are known as ___________

A compound proposition that is always ___________ is called a tautology.

In proving √5 as irrational, we begin with assumption √5 is rational in which type of proof?

A compound proposition that is always ___________ is called a contradiction.

(A ∨ ¬A) ∨ (q ∨ T) is a __________

(A ∨ F) ∨ (A ∨ T) is always _________

A ∧ ¬(A ∨ (A ∧ T)) is always __________

¬ (A ∨ q) ∧ (A ∧ q) is a ___________

Let R (x) denote the statement “x > 2.” What is the truth value of the quantification ∃xR(x), having domain as real numbers?

A → (A ∨ q) is a __________

Let P(x) denote the statement “x = x + 7.” What is the truth value of the quantification ∃xP(x), where the domain consists of all real numbers?

The statement,” Every comedian is funny” where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people.

The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people.

Let domain of m includes all students, P (m) be the statement “m spends more than 2 hours in playing polo”. Express ∀m ¬P (m) quantification in English.

”Everyone wants to learn cosmology.” This argument may be true for which domains?

Let Q(x, y) denote “M + A = 0.” What is the truth value of the quantifications ∃A∀M Q(M, A).

Determine the truth value of ∀n(n + 1 > n) if the domain consists of all real numbers.

45.	The statement,” Every comedian is funny” where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people.
A.	∃x(C(x) ∧ F (x))
B.	∀x(C(x) ∧ F (x))
C.	∃x(C(x) → F (x))
D.	∀x(C(x) → F (x))
Answer» E.

46.	The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people.
A.	∀x (F (x) → P (x))
B.	∀x (F (x) ∧ P (x))
C.	∃x (F (x) ∧ P (x))
D.	∃x (F (x) → P (x))
Answer» D. ∃x (F (x) → P (x))