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## Discrete Mathematics MCQ Set 1

1. Which of the following bits is the negation of the bits “010110”?
a) 111001
b) 001001
c) 101001
d) 111111

Answer: c [Reason:] Flip each of the bit to get the negation of the required string.

2. Which of the following option is suitable, if A is “10110110”, B is”11100000”and C is”10100000”?
a) C=A or B
b) C=~A
c) C=~B
d) C=A and B

Answer: d [Reason:] Output of and is 1 when both other inputs are one.

3. How many bits string of length 4 are possible such that they contains 2 ones and 2 zeroes?
a) 4
b) 2
c) 5
d) 6

Answer: d [Reason:] The strings are {0011, 0110, 1001, 1100, 1010 and 0101}.

4. If a bit string contains {0, 1} only, having length 5 has no more than 2 ones in it. Then how many such bit strings are possible?
a) 14
b) 12
c) 15
d) 16

Answer: d [Reason:] The total strings are 1(having no one in it) +5(having 1 one in it) +10 (having 2 ones in it) = 16.

5. If A is “001100” and B is “010101” then A (Ex-or) B is
a) 000000
b) 111111
c) 001101
d) 011001

Answer: d [Reason:] In Ex-or if both the inputs are same then output is 0 otherwise 1.

6. The Ex-nor of this string “01010101”with “11111111” is
a) 10101010
b) 00110100
c) 01010101
d) 10101001

Answer: c [Reason:] In Ex-nor if both the inputs are same then output is 1 otherwise 0.

7. The one’s complement of this string “01010100” is
a) 10101010
b) 00110101
c) 10101011
d) 10101001

Answer: c [Reason:] Negate every bit in one’s complement.

8. The 2’s complement of this string “01010100” is
a) 10101010
b) 00110100
c) 10101100
d) 10101001

Answer: c [Reason:] In two’s complement negate every bit from left until the first one from right is encountered.

9. If in a bits string of {0,1} ,of length 4,such that no two ones are together. Then total number of such possible strings are?
a) 1
b) 5
c) 7
d) 4

Answer: c [Reason:] Strings can be {1001, 1010, 0101, 1000, 0100, 0010, 0001}.

10. Let A : “010101” ,B=? ,If { A (Ex-or) B } is a resultant string of all ones then which of the following statement regarding B is correct
a) B is negation of A
b) B is 101010
c) {A (and) B} is a resultant string having all zeroes
d) All of the mentioned

Answer: d [Reason:] In Ex-or both if both the inputs are same then output is 0 otherwise 1.

## Discrete Mathematics MCQ Set 2

1. If there are ‘M’ switches in series numbered from 1, 2, …, M. For circuit to be complete and bulb to glow which of the following is necessary
a) 1∧ 2∧ 3 ∧ … ∧M should be on
b) 1∧ 2∧ 3 ∧ … ∧M should be off
c) 1 v 2 v 3 v … v M should be on
d) None of the mentioned

Answer: a [Reason:] All should be on in-order to complete the circuit.

2. If there are ‘M’ switches in parallel numbered from 1, 2, …, M. For circuit to be complete and bulb to glow which of the following is necessary
a) 1∧ 2∧ 3 ∧ … ∧M should be on
b) 1∧ 2∧ 3 ∧ … ∧M should be off
c) 1 v 2 v 3 v … v M should be on
d) None of the mentioned

Answer: c [Reason:] Anyone should be on in-order to complete the circuit.

3. In the circuit shown the lamp will be glowing if a) P: True ,Q: False
b) P: True, Q: True
c) P: False, Q: False
d) None of the mentioned

Answer: a [Reason:] The circuit will be complete if P is true and Q is false.

4. In this circuit shown the lamp will be glowing if a) P: True ,Q: True, R: False
b) P: True, Q: True, R: True
c) P: False, Q: False, R: True
d) None of the mentioned

Answer: c [Reason:] The circuit will be complete if R is true and Q is false or P is false.

5. Which statement should be true in order for lamp to glow ? a) (R ∧ (~(P ∧ Q))
b) P∧R∧Q
c) P ∧ (Q ∧ ~R)
d) None of the mentioned

Answer: a [Reason:] The circuit will be complete if R is true and Q is false or P is false.

6. If it is given that switch R is closed and Q is closed then lamp will glow if a) P: Open , S: Closed
b) P: Open , S: Open
c) P: Closed , S: Closed
d) None of the mentioned

Answer: a [Reason:] The circuit will be complete if (~P) is true and S is true.

7. The circuit depend on which switch/switches state to be complete? a) P
b) Q
c) Both P and Q
d) None of the mentioned

Answer: a [Reason:] The circuit will be complete if (P) is true , Q v ~Q will always be true.

8. The ten switches A,B,C,D,E,F,G,H,N,M are placed in the given circuit (all are open at given time).If you close one switch you need to pay 1 unit cost?
What is the cost you need to pay to glow this Lamp? a) 1 units
b) 2 units
c) 3 units
d) 4 units

Answer: a [Reason:] This can be achieved by turning one of switches N or M on.

9. If in a for it to be complete it is necessary for switch A to be closed and either of switch B or C to be closed, then which can be true?
a) Switch A should in parallel with B and C is series to them
b) Switch A should be in series with parallel circuit of B and C
c) All of the mentioned
d) None of the mentioned

Answer: b [Reason:] Switch A is in series and since there is ‘or’ between B and C therefore they must be in parallel.

10. The given circuit can work if the switches P and Q be a) P: True ,Q: False
b) P: True, Q: True
c) P: False, Q: False
d) None of the mentioned

Answer: d [Reason:] As no combination of switch can make this circuit operative. Since it is not possible for P and ~P to be simultaneously true.

## Discrete Mathematics MCQ Set 3

1. The compound propositions p and q are called logically equivalent if ________ is a tautology.
a) p ↔ q
b) p → q
c) ¬ (p ∨ q)
d) ¬p ∨ ¬q

Answer: a [Reason:] Definition of logical equivalence.

2. p → q is logically equivalent to:
a) ¬p ∨ ¬q
b) p ∨ ¬q
c) ¬p ∨ q
d) ¬p ∧ q

Answer: c [Reason:] (p → q) ↔ (¬p ∨ q) is tautology.

3. p ∨ q is logically equivalent to:
a) ¬q → ¬p
b) q → p
c) ¬p → ¬q
d) ¬p → q

Answer: d [Reason:] (p ∨ q) ↔ (¬p → q) is tautology.

4. ¬ (p ↔ q) is logically equivalent to:
a) q↔p
b) p↔¬q
c) ¬p↔¬q
d) ¬q↔¬p

Answer: b [Reason:] ¬(p↔q)↔(p↔¬q) is tautology.

5. p ∧ q is logically equivalent to:
a) ¬ (p → ¬q)
b) (p → ¬q)
c) (¬p → ¬q)
d) (¬p → q)

Answer: a [Reason:] (p ∧ q) ↔ (¬(p → ¬q)) is tautology.

6. Which of the following statement is correct?
a) p ∨ q ≡ q ∨ p
b) ¬(p ∧ q) ≡ ¬p ∨ ¬q
c) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
d) All of mentioned

Answer: d [Reason:] Verify using truth table, all are correct.

7. p ↔ q is logically equivalent to:
a) (p → q) → (q → p)
b) (p → q) ∨ (q → p)
c) (p → q) ∧ (q → p)
d) (p ∧ q) → (q ∧ p)

Answer: c [Reason:] (p ↔ q) ↔ ((p → q) ∧ (q → p)) is tautology.

8. (p → q) ∧ (p → r) is logically equivalent to:
a) p → (q ∧ r)
b) p → (q ∨ r)
c) p ∧ (q ∨ r)
d) p ∨ (q ∧ r)

Answer: a [Reason:] ((p → q) ∧ (p → r)) ↔ (p → (q ∧ r)) is tautology.

9. (p → r) ∨ (q → r) is logically equivalent to:
a) (p ∧ q) ∨ r
b) (p ∨ q) → r
c) (p ∧ q) → r
d) (p → q) → r

Answer: c [Reason:] ((p → r) ∨ (q → r)) ↔ ((p ∧ q) → r) is tautology.

10. ¬ (p ↔ q) is logically equivalent to:
a) p ↔ ¬q
b) ¬p ↔ q
c) ¬p ↔ ¬q
d) ¬q ↔ ¬p

Answer: a [Reason:] (¬ (p ↔ q)) ↔ (p ↔ ¬q) is tautology.

## Discrete Mathematics MCQ Set 4

1. If a matrix A = [A11 A12 ⋯ A1n A21 A2n ⋮ ⋮ An1 An2 ⋯ Ann ], order(nxn) Aii = 1, Aij = 0 for i ≠ j. Then that matrix is know as
a) Identity matrix
b) Null matrix
c) Singular matrix
d) None of the mentioned

Answer: a [Reason:] In unit matrix all diagonal elements are 1 and all other 0.

2. A symmetric matrix is a one in which
a) All diagonal elements are zero
b) All diagonal elements are 1
c) A = AT
d) A = -AT

Answer: c [Reason:] For symmetric matrices, matrix remains same even after transpose.

3. An anti-symmetric matrix is a one in which
a) All diagonal elements are zero
b) All diagonal elements are 1
c) A = AT
d) A = -AT

Answer: d [Reason:] Foran anti-symmetric matrix, matrix changes it sign after transpose.

4. A matrix having one row and many columns is known as?
a) Row matrix
b) Column matrix
c) Diagonal matrix
4) None of the mentioned

Answer: a [Reason:] In row matrix there is only one row.

5. A matrix having many rows and one column is known as?
a) Row matrix
b) Column matrix
c) Diagonal matrix
4) None of the mentioned

Answer: b [Reason:] In column matrix there is only one column.

6. The trace of the matrix is defined as:
a) Sum of all the elements of the matrix
b) Sum of all the elements of leading diagonal of matrix
c) Sum of all non-zero elements of matrix
d) None of the mentioned

Answer: b [Reason:] Trace is the sum of the elements of leading diagonal of matrix.

7. A square matrix A = [aij ]nxn, if aij = 0 for i > j then that matrix is known as:
a) Upper triangular matrix
b) Lower triangular matrix
c) Unit matrix
d) Null matrix

Answer: a [Reason:] In upper triangular matrix A = [ aij ]nxn, if aij = 0 for i > j.

8. A square matrix A = [aij ]nxn, if aij = 0 for i < j then that matrix is known as:
a) Upper triangular matrix
b) Lower triangular matrix
c) Unit matrix
d) Null matrix

Answer: b [Reason:] In lower triangular matrix A = [aij ]nxn, if aij = 0 for i < j.

9. Two matrix can be added if:
a) rows of both the matrices are same
b) columns of both the matrices are same
c) both rows and columns of both the matrices are same
d) number of rows of first matrix should be equal to number of column of second

Answer: c [Reason:] Order of two matrices must be same.

10. For matrix A if AAT = I, I is identity matrix then A is :
a) Orthagonal matrix
b) Nilpotent matrix
c) Idempotent matrix
d) None of the mentioned

Answer: a [Reason:] For orthagonal matrices AAT = I = AT A.

## Discrete Mathematics MCQ Set 5

1. Domain of a function is :
a) the maximal set of numbers for which a function is defined
b) the maximal set of numbers which a function can take values
c) it is set of natural numbers for which a function is defined
d) none of the mentioned

Answer: a [Reason:] Domain is the set of all the numbers on which a function is defined.It may be real as well.

2. What is domain of function f(x)= x1/2 ?
a) (2, ∞)
b) (-∞, 1)
c) [0, ∞)
d) None of the mentioned

Answer: c [Reason:] A square root function is not defined for negative real numbers.

3. Range of a function is :
a) the maximal set of numbers for which a function is defined
b) the maximal set of numbers which a function can take values
c) it is set of natural numbers for which a function is defined
d) none of the mentioned

Answer: b [Reason:] Range is the set of all values which a function may take.

4. What is domain of function f(x) = x-1 for it to be defined everywhere on domain?
a) (2, ∞)
b) (-∞, ∞) – {0}
c) [0, ∞)
d) None of the mentioned

Answer: b [Reason:] Function x-1 is not defined for x=0,otherwise it defined for every real number.

5. State whether the given statement is true or false
The range of function f(x) = sin(x) is (-∞, ∞).
a) True
b) False

Answer: b [Reason:] A sine function takes values between -1 and 1,thus range is [-1, 1].

6. State whether the given statement is true or false
Codomain is the subset of range.
a) True
b) False

Answer: b [Reason:] Range is the subset of codomain, that is every value in range is in codomain but vice-versa it is not true.

7. What is range of function f(x) = x-1 which is defined everywhere on its domain?
a) (-∞, ∞)
b) (-∞, ∞) – {0}
c) [0, ∞)
d) None of the mentioned

Answer: a [Reason:] Function x-1 may take any real number hence it’s range is all real numbers.

8. If f(x) = 2x then range of the function is :
a) (-∞, ∞)
b) (-∞, ∞) – {0}
c) (0, ∞)
d) None of the mentioned

Answer: c [Reason:] The function cannot take negative values,hence range is (0, ∞).

9. If f(x) = x2 + 4 then range of f(x) is given by
a) [4, ∞)
b) (-∞, ∞) – {0}
c) (0, ∞)
d) None of the mentioned