Discrete Mathematics MCQ Set 1
1. Let Q(x, y) denote “M + A = 0.” What is the truth value of the quantifications ∃A∀M Q(M, A)
a) True
b) False
Answer
Answer: b [Reason:] For each A there exist only one M, because there is no real number A such that M + A = 0 for all real numbers M.
2. Translate ∀x∃y(x < y) in English, considering domain as real number for both the variable.
a) For all real number x there exists a real number y such that x is less than y
b) For every real number y there exists a real number x such that x is less than y
c) For some real number x there exists a real number y such that x is less than y
d) For each and every real number x and y such that x is less than y
Answer
Answer: a [Reason:] Statement is x is less than y. Quantifier used are for each x, there exist a y.
3. “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
Answer: d [Reason:] For every negative real number x and y, the product of these integer is positive.
4. 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
Answer
Answer: b [Reason:] There exist no integer for which x+4=x-4.
5. 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
Answer: a [Reason:] Joy is loved by all the people in the world.
6. 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.
a) All students does not like South Indian dishes.
b) Amit does not like South Indian people.
c) Amit does not like South Indian dishes.
d) Amit does not like some dishes.
Answer
Answer: d [Reason:] Negation of the statement Amit like South Indian dishes.
7. 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
Answer: b [Reason:] For every real number x, difference with itself is always zero.
8. 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.”
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
b) ∃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
c) ∀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
d) ∃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
Answer
Answer: a [Reason:] For some x pupil, there exist a course in Discrete Maths such that x has taken y.
9. 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
Answer
Answer: b [Reason:] The equation does not satisfy any value of m and n in the domain consist of integers.
10. Find a counterexample 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 a and b
d) Does not have any counter example
Answer
Answer: c [Reason:] Putting x=-1, y=17; -17>17 which is wrong. Putting x=-2, y=8; -16>8 which is wrong.
Discrete Mathematics MCQ Set 2
1. If there exist an integer x such that x2 ≡ q (mod n). then q is called:
a) Quadratic Residue
b) Linear Residue
c) Pseudoprime
d) None of the mentioned
Answer
Answer: a [Reason:] q is called quadratic residue if it is congurent to a perfect square modulo n.
2. If there exist no integer x such that x2 ≡ q (mod n). then q is called:
a) Quadratic Residue
b) Quadratic Nonresidue
c) Pseudoprime
d) None of the mentioned
Answer
Answer: b [Reason:] q is called quadratic nonresidue if it is not congurent to a perfect square modulo n.
3. The Fermat’s little theorem for odd prime p and coprime number a,is:
a) ap-1 ≡ 1 (mod p)
b) ap-1 ≡ 7 (mod p)
c) ap(2)-1 ≡ 1 (mod p)
d) None of the mentioned
Answer
Answer: a [Reason:] According to Fermat’s little theorem ap-1 ≡ 1 (mod p).
4. State whether the given statement is true or false
5 is quardratic non-residue of 7.
a) True
b) False
Answer
Answer: a [Reason:] Since there exist no number which gives 5 modullo 7 when squared.
5. State whether the given statement is true or false
4 is quardratic residue of 7.
a) True
b) False
Answer
Answer: a [Reason:] Since 25 ≡ 4(mod)7 , 4 is quardratic residue of 7.
6. State whether the given statement is true or false
8 is quardratic residue of 17.
a) True
b) False
Answer
Answer: a [Reason:] Since 25 ≡ 8(mod)17.
7. State whether the given statement is true or false
8 is quardratic residue of 11 .
a) True
b) False
Answer
Answer: b [Reason:] Since x2 ≡ 8(mod)17 has no solutions.
8. Which of the following is a quardratic residue of 11?
a) 4
b) 5
c) 9
d) All of the mentioned
Answer
Answer: d [Reason:] Since 4, 16, 32 satisfies the criteria, all are quardratic residue of 11.
9. A pseudo prime number is
a) is a probable prime and is not a prime number
b) is a prime number
c) does not share any property with prime number
d) none of the mentioned
Answer
Answer: a [Reason:] A pseudo prime number is an integer that shares a property common to all prime number and is not a prime number.
10. Pseudo prime are classified based on property which they satisfy,which of the following are classes of pseudoprimes:
a) Fermat pseudoprime
b) Fibonacci pseudoprime
c) Euler pseudoprime
d) All of the mentioned
Answer
Answer: d [Reason:] Fermat pseudoprime, Fibonacci pseudoprime, Euler pseudoprime are differnet classes of pseudoprimes.
Discrete Mathematics MCQ Set 3
1. Let A and B be two matrices of same order ,then state whether the given statement is true or false:
A + B = B + A
a) True
b) False
Answer
Answer: a [Reason:] Matrix addition is commutative.
2. Let A and B be two matrices of same order, then state whether the given statement is true or false:
AB = BA
a) True
b) False
Answer
Answer: b [Reason:] Matrix multiplication is not commutative.
3. Let A order(axb) and Border(cxd) be two matrices, then for AB to exist, correct relation is given by:
a) a = d
b) b = c
c) a = b
d) c = d
Answer
Answer: b [Reason:] Matrix multiplication exists only when column of first matrix is same as rows of second i.e b = c.
4. Let A order(axb) and Border(cxd) be two matrices, then if AB exists, the order of AB is:
a) axd
b) bxc
c) axb
d) cxd
Answer
Answer: a [Reason:] Matrix multiplication exists only when column of first matrix is same as rows of second i.e b = c also resultant matrix will have number of rows equal to first matrix and column equal to second matrix.
5. Let A=[aij ] be an mxn matrix and k be a scalar then kA is equal to :
a) [kaij ]mxn
b) [aij/k ]mxn
c) [k2 aij ]mxn
d) None of the mentioned
Answer
Answer: a [Reason:] The scalar is multiplied with each of the element of matrix A.
6. State True or False:
The matrix multiplication is distrbutive over matrix addition.
a) True
b) False
Answer
Answer: a [Reason:] For matrix A, B, C, A(B+C) = AB + AC.
7. If for a square matrix A, A2 = A then such a matrix is known as:
a) Idempotent matrix
b) Orthagonal matrix
c) Null matrix
d) None of the mentioned
Answer
Answer: a [Reason:] A sqaure matrix is called an Idempotent matrix, if A2 = A.
8. State whether the given statement is True or False.
For matrix A, B.(A+B)T = AT + BT and (AB)T = ATBT if the orders of matrices are appropriate.
a) True
b) False
Answer
Answer: b [Reason:] (A+B)T = AT + BT is correct but (AB)T = BTAT(reversal law).
9. For matrix A, B if A – B = O, where O is a null matrix then
a) A = O
b) B = O
c) A = B
d) None of the mentioned
Answer
Answer: c [Reason:] If subtraction of B from A results in null matrix this means that A is equivalent to B.
10. All the diagonal elements of a skew-symmetric matrix is:
a) 0
b) 1
c) 2
d) Any integer
Answer
Answer: a [Reason:]Since for a skew symmetric matrix aij = -aij, this implies all diagonal elements should be zero.
Discrete Mathematics MCQ Set 4
1. The number of factors of a prime numbers are:
a) 2
b) 3
c) Depends on the prime number
d) None of the mentioned
Answer
Answer: a [Reason:] A prime number is only divisible by 1 and itself.
2. The number ‘ 1’ is :
a) Prime number
b) Composite number
c) Neither Prime nor Composite
d) None of the mentioned
Answer
Answer: c [Reason:] 1 is neither prime number nor composite.
3. State whether True or False
All prime numbers are odd.
a) True
b) False
Answer
Answer: b [Reason:] 2 is even as well as prime.
4. State whether True or False
3 is the smallest prime number possible.
a) True
b) False
Answer
Answer: b [Reason:] 2 is also a prime number.
5. How may prime numbers are there between 1 to 20.
a) 5
b) 6
c) 7
4) None of the mentioned
Answer
Answer: d [Reason:] The prime numbers between 1 to 20 are 2, 3, 5, 7, 11, 13, 17, 19.
6. State whether the given statement is true or false
There are finite number of prime numbers.
a) True
b) False
Answer
Answer: b [Reason:] There are infinite numbers of primes.
7. Sum of two different prime number is a:
a) Prime number
b) Composite number
c) Either Prime or Composite
d) None of the mentioned
Answer
Answer: c [Reason:] Eg:- 2 + 3 = 5 a prime, 3 + 7 = 10 a composite.
8. Difference of two distinct prime numbers is ?
a) Odd and prime
b) Even and composite
c) None of the mentioned
d) All of the mentioned
Answer
Answer: c [Reason:] 3 – 2 = 1 is neither prime nor composite.
9. If a, b, c, d are distinct prime numbers with a as smallest prime then a * b * c * d is a:
a) Odd number
b) Even number
c) Prime number
d) None of the mentioned.
Answer
Answer: b [Reason:] Since a is 2, 2 * b * c * d = Even number.
10. If a, b are two distinct prime number than highest common factor of a, b is
a) 2
b) 0
c) 1
d) ab
Answer
Answer: c [Reason:] HCF of two prime numbers is 1.
Discrete Mathematics MCQ Set 5
1. The determinant of identity matrix is :
a) 1
b) 0
c) Depends on the matrix
d) None of the mentioned
Answer
Answer: a [Reason:] In identity matrix aii = 1, and all other elements = 0, hence the determinant is 1.
2. If determinant of a matrix A is Zero than:
a) A is a Singular matrix
b) A is a non-Singular matrix
c) Can’t say
d) None of the mentioned
Answer
Answer: a [Reason:] Determinant of singular matrices are zero.
3. For a skew symmetric even ordered matrix A of integers, which of the following will not hold true:
a) det(A) = 9
b) det(A) = 81
c) det(A) = 7
d) det(A) = 4
Answer
Answer: c [Reason:] Determinant of a skew symmetric even ordered matrix A is a perfect square.
4. For a skew symmetric odd ordered matrix A of integers, which of the following will hold true:
a) det(A) = 9
b) det(A) = 81
c) det(A) = 0
d) det(A) = 4
Answer
Answer: c [Reason:] Determinant of a skew symmetric odd ordered matrix A is always 0 .
5. Let A = [kaij ]nxn, B = [aij ]nxn, be an nxn matrices and k be a scalar then det(A) is equal to:
a) Kdet(B)
b) Kndet(B)
c) K3det(b)
4) None of the mentioned
Answer
Answer: b [Reason:] The scalar is multiplied with each of the element of matrix A then determinant is multiplied, the number of row times to the scalar i.e. Kndet(B).
6. State True or False:
The Inverse exist only for non-singular matrices.
a) True
b) False
Answer
Answer: a [Reason:] Since for singular matrix det(A)=0.Hence Inverse does not exist.
7. State True or False:
If for a square matrix A and B,null matrix O, AB =O implies BA=O:
a) True
b) False
Answer
Answer: b [Reason:] Let A = [0 1 0 0 ], B = [1 0 0 0 ]AB=O and BA is not equal to O.
8. State whether the given statement is True or False.
If for a square matrix A and B,null matrix O, AB =O implies A=O and B=O.
a) True
b) False
Answer
Answer: b [Reason:] Let A = [0 1 0 0 ], B = [1 0 0 0 ]AB=O and B, A is not equal to O.
9. Let A be a nilpotent matrix of order n then
a) An = O
b) nA = O
c) A = nI, I is Identity matrix
d) None of the mentioned
Answer
Answer: a [Reason:] n is the smallest possible number such that An = O.
10. Which of the following property of matrix multiplication is correct:
a) Multiplication is not commutative in genral
b) Multiplication is associative
c) Multiplication is distributive over addition
d) All of the mentioned
Answer
Answer: d [Reason:] Matrix multiplication is associative, distributive, but not commutative.
