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## Discrete Mathematics MCQ Set 1

1. Which of the following statements is the negation of the statements “4 is odd or -9 is positive”?
a) 4 is even or -9 is not negative
b) 4 is odd or -9 is not negative
c) 4 is even and -9 is negative
d) 4 is odd and -9 is not negative

Answer: c [Reason:] Using De Morgan’s Law ~(A V B) <-> ~A ∧ ~B.

2. Which of the following represents: ~A (negation of A) if A stands for “I like badminton but hate maths”?
a) I hate badminton and maths
b) I do not like badminton or maths
c) I dislike badminton but love maths
d) I hate badminton or like maths

Answer: d [Reason:] De Morgan’s Law ~ (A ∧ B) <-> ~A V ~B.

3. The compound statement A v ~(A ∧ B) is always
a) True
b) False

Answer: a [Reason:] Applying De-Morgan’s law we get A v ~ A Ξ Tautology.

4. Which of the following are De-Morgan’s law
1) P ∧ (Q v R) Ξ ( P ∧ Q ) v ( P ∧ R )
2) ~(P ∧ R) Ξ ~P v ~R , ~(P v R) Ξ ~P ∧ ~R
3) P v ~P Ξ True , P ∧ ~P Ξ False
4) None of the mentioned

Answer: b [Reason:] Definition of De –Morgan’s Law.

5. What is the dual of (A ∧ B) v ( C ∧ D) ?
a) (A V B) v ( C v D)
b) (A V B) ^ ( C v D)
c) (A V B) v ( C ∧ D)
d) (A ∧ B) v ( C v D)

Answer: b [Reason:] In dual ∧ is replaced by v and vice – versa.

6. ~ A v ~ B is logically equivalent to
a) ~ A → ~ B
b) ~ A ∧ ~ B
c) A → ~B
d) B V A

Answer: c [Reason:] By identity A → B Ξ ~A V B.

7. Negation of statement (A ∧ B) → (B ∧ C)
a) (A ∧ B) →(~B ∧ ~C)
b) ~(A ∧ B) v ( B v C)
c) ~(A →B) →(~B ∧ C)
d) None of the mentioned

Answer: a [Reason:] ~(A →B) Ξ A ∧ ~B using this we can easily fetch the answer.

8. Which of the following satisfies commutative law?
a) ∧
b) v
c) <->
d) All of the mentioned

Answer: d [Reason:] All of them satisfies commutative law.

9. If the truth value of A v B is true, then truth value of ~A ∧ B can be
a) True if A is false
b) False if A is false
c) False if B is true and A is false
d) None of the mentioned

Answer: a [Reason:] If A is false then both the condition are obeyed.

10. If P is always against the testimony of Q ,then the compound statement P→(P v ~Q) is a
a) Tautology
c) Contingency
d) None of the mentioned

Answer: a [Reason:] Since either hypothesis is false or both (hypothesis as well as conclusion) are true.

## Discrete Mathematics MCQ Set 2

1. For an inverse to exist it is necessary that a function should be :
a) injection
b) bijection
c) surjection
d) none of the mentioned

Answer: b [Reason:] Inverse exist only for those functions which are one one and onto.

2. If f(x) = y then f -1(y) is equal to :
a) y
b) x
c) x2
d) none of the mentioned

Answer: b [Reason:] On giving inverse, image the function returns preimage thus f -1 (y) = x.

3. A function f(x) is defined from A to B then f -1 is defined :
a) from A to B
b) from B to A
c) depends on the inverse of function
d) none of the mentioned

Answer: b [Reason:] Inverse associate each element in B with corresponding element in A.

4. If f is a function defined from R to R , is given by f(x) = 3x – 5 then f –1(x) is given by:
a) 1/(3x-5)
b) (x+5)/3
c) does not exist since it is not a bijection
d) none of the mentioned

Answer: b [Reason:] y = 3x-5, x = (y+5)/3, f -1(x) = (x+5)/3.

5. State whether the given statement is true or false
For some bijective function inverse of that function is not bijective.
a) True
b) False

Answer: b [Reason:] If f(x) is a bijection than f -1(x) is also a bijection.

6. State whether the given statement is true or false
f(x) is a bijection than f -1(x) is a mirror image of f(x) around y = x.
a) True
b) False

Answer: a [Reason:] Inverse of a function is the mirror image of function in line y = x.

7. If f is a function defined from R to R , is given by f(x) = x2 then f –1(x) is given by:
a) 1/(3x-5)
b) (x+5)/3
c) does not exist since it is not a bijection
d) none of the mentioned

Answer: c [Reason:] It is not a one one function hence Inverse does not exist.

8. For any function fof -1(x) is equal to ?
a) x
b) 1
c) x2
d) none of the mentioned

Answer: a [Reason:]Compostion of a function with its inverse gives x.

9. The solution to f(x) = f -1(x) are :
a) no solutions in any case
b) same as solution to f(x) = x
c) infinite number of solution for every case
d) none of the mentioned

Answer: b [Reason:] Inverse of a function is the mirror image of function in line y = x.

10. State True or False.
Let f(x) = x then number of solution to f(x) = f -1(x) is zero.
a) True
b) False

Answer: b [Reason:] Since inverse of a function is the mirror image of function in line y = x, therefore in this case infinte solution will exist.

## Discrete Mathematics MCQ Set 3

1. A compound proposition that is always ___________ is called a tautology.
a) True
b) False

Answer: a [Reason:] Tautology is always true.

2. A compound proposition that is always ___________ is called a contradiction.
a) True
b) False

3. If A is any statement, then which of the following is a tautology?
a) A ∧ F
b) A ∨ F
c) A ∨ ¬A
d) A ∧ T

Answer: c [Reason:] A ∨ ¬A is always true.

4. If A is any statement, then which of the following is not a contradiction?
a) A ∨ ¬A
b) A ∨ F
c) A ∧ F
d) None of mentioned.

Answer: b [Reason:] A ∨ F is not always false.

5. A compound proposition that is neither a tautology nor a contradiction is called a ___________
a) Contingency
b) Equivalence
c) Condition
d) Inference

Answer: a [Reason:] Definition of contingency.

6. ¬ (A ∨ q) ∧ (A ∧ q) is a ___________
a) Tautology
c) Contingency
d) None of the mentioned

Answer: b [Reason:] ≡ (¬A ∧ ¬q) ∧ (A ∧ q) ≡ (¬A ∧ A) ∧ (¬q ∧ q) ≡ F ∧ F ≡ F.

7. (A ∨ ¬A) ∨ (q ∨ T) is a __________
a) Tautology
c) Contingency
d) None of the mentioned

Answer: a [Reason:] ≡ (A ∨ ¬A) ∨ (q ∨ T) ≡ T ∨ T ≡ T.

8. A ∧ ¬(A ∨ (A ∧ T)) is always __________
a) True
b) False

Answer: b [Reason:] ≡ A ∧ ¬ (A ∨ (A ∧ T)) ≡ A ∧ ¬(A ∨ A) ≡ A ∧ ¬A ≡ F.

9. (A ∨ F) ∨ (A ∨ T) is always _________
a) True
b) False

Answer: a [Reason:] ≡ (A ∨ F) ∨ (A ∨ T) ≡ A ∨ T ≡ T.

10. A → (A ∨ q) is a __________
a) Tautology
c) Contingency
d) None of the mentioned

Answer: a [Reason:] ≡ A → (A ∨ q) ≡ ¬A ∨ (A ∨ q) ≡ (A ∨ ¬A) ∨ q ≡ T ∨ q ≡ T.

## Discrete Mathematics MCQ Set 4

1. A floor function map a real number to :
a) smallest previous integer
b) greatest previous integer
c) smallest following integer
d) none of the mentioned

Answer: b [Reason:] Floor function f(x) is the largest integer not greater than x.

2. A ceil function map a real number to :
a) smallest previous integer
b) greatest previous integer
c) smallest following integer
d) none of the mentioned

Answer: c [Reason:] Ceil function f(x) is the smallest integer not less than x.

3. A function f(x) is defined as f(x) = x – [x], where [.] represents GIF then:
a) f(x) will be intergral part of x
b) f(x) will be fractional part of x
c) f(x) will always be 0
d) none of the mentioned

Answer: b [Reason:] A integral part of a number is subtracted from that number we are left with the fractional part of that number.

4. Floor(2.4) + Ceil(2.9) is equal to :
a) 4
b) 6
c) 5
d) none of the mentioned

Answer: c [Reason:] Floor(2.4) = 2, Ceil(2.9) = 3, 2 + 3 = 5.

5. State whether the given statement is true or false
For some integer n such that x < n < x + 1, ceil(x) < n .
a) True
b) False

Answer: b [Reason:] If x < n < x + 1 then ceil(x) = n.

6. State whether the given statement is true or false
For some number x, Floor(x) <= x <= Ceil(x).
a) True
b) False

Answer: a [Reason:] Floor function f(x) is the largest integer not greater than x and ceil function f(x) is the smallest integer not less than x.

7. If x, and y are positive numbers both are less than one, then maximum value of floor(x + y) is?
a) 0
b) 1
c) 2
d) -1

Answer: b [Reason:] Since x < 1 and y < 1 this implies x + y < 2 which means maximium value of floor(x + y) is 1.

8. If x, and y are positive numbers both are less than one ,then maximum value of ceil(x + y) is?
a) 0
b) 1
c) 2
d) -1

Answer: c [Reason:] Since x < 1 and y < 1 this implies x + y < 2 which means maximum value of ceil(x + y) is 2.

9. If X = Floor(X) = Ceil(X) then :
a) X is a fractional number
b) X is a Integer
c) X is less than 1
d) none of the mentioned

Answer: b [Reason:] Only in case of integers X = Floor(X) = Ceil(X) holds good.

10.State True or False.
Let n be some integer greater than 1,then floor((n-1)/n) is 1.
a) True
b) False

Answer: b [Reason:] Since (n-1)/n will always be less than one thus f floor((n-1)/n) is 0.

## Discrete Mathematics MCQ Set 5

1. To measure Time complexity of an algorithm Big O notation is used which:
a) describes limiting behaviour of the function
b) characterises a function based on growth of function
c) upper bound on growth rate of the function
d) all of the mentioned

Answer: d [Reason:] Big O notation describes limiting behaviour, and also gives upper bound on growth rate of a function.

2. If for an algorithm time complexity is given by O(1) then complexityof it is:
a) constant
b) polynomial
c) exponential
d) none of the mentioned

Answer: a [Reason:] The growth rate of that function will be constant.

3. If for an algorithm time complexity is given by O(log2n) then complexity will:
a) constant
b) polynomial
c) exponential
d) none of the mentioned

Answer: d [Reason:] The growth rate of that function will be logarithmic therefore complexity will be logarithmic.

4. If for an algorithm time complexity is given by O(n) then complexityof it is:
a) constant
b) linear
c) exponential
d) none of the mentioned

Answer: b [Reason:] The growth rate of that function will be linear.

5. If for an algorithm time complexity is given by O(n2) then complexity will:
a) constant
b) quardratic
c) exponential
d) none of the mentioned

Answer: b [Reason:] The growth rate of that function will be quadratic therefore complexity will be quardratic.

6. If for an algorithm time complexity is given by O((32)n) then complexity will:
a) constant
b) quardratic
c) exponential
d) none of the mentioned

Answer: c [Reason:] The growth rate of that function will be exponential therefore complexity will be exponential.

7. The time complexity of binary search is given by:
a) constant
b) quardratic
c) exponential
d) none of the mentioned

Answer: d [Reason:] It is O(log2n), therefore complexity will be logarithmic.

8. The time complexity of linear search is given by:
a) O(log2n)
b) O(1)
c) exponential
d) none of the mentioned

Answer: d [Reason:] It is O(n), therefore complexity will be linear.

9. Which algorithm is better for sorting between bubble sort and quicksort?
a) bubble sort
b) quick sort
c) both are equally good
d) none of the mentioned