# Number System & Logics

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

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

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

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

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

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

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

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

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

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

11.Which of the following statement is a proposition?
a) Get me a glass of milkshake
b) God bless you!
c) What is the time now?
d) The only odd prime number is 2

12.The truth value of ‘4+3=7 or 5 is not prime’.
a) False
b) True

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13.Which of the following option is true?
a) If the Sun is a planet, elephants will fly
b) 3 +2 = 8 if 5-2 = 7
c) 1 > 3 and 3 is a positive integer
d) -2 > 3 or 3 is a negative integer

Explanation: Hypothesis is false, thus the whole statement is true.

14.What is the value of x after this statement, assuming the initial value of x is 5?
‘If x equals to one then x=x+2 else x=0’.
a) 1
b) 3
c) 0
d) 2

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15.Let P: I am in Bangalore.; Q: I love cricket.; then q -> p(q implies p) is?
a) If I love cricket then I am in Bangalore
b) If I am in Bangalore then I love cricket
c) I am not in Bangalore
d) I love cricket

16.Let P: If Sahil bowls, Saurabh hits a century.; Q: If Raju bowls, Sahil gets out on first ball. Now if P is true and Q is false then which of the following can be true?
a) Raju bowled and Sahil got out on first ball
b) Raju did not bowled
c) Sahil bowled and Saurabh hits a century
d) Sahil bowled and Saurabh got out

17.The truth value ‘9 is prime then 3 is even’.
a) False
b) True

18.Let P: I am in Delhi.; Q: Delhi is clean.; then q ^ p(q and p) is?
a) Delhi is clean and I am in Delhi
b) Delhi is not clean or I am in Delhi
c) I am in Delhi and Delhi is not clean
d) Delhi is clean but I am in Mumbai

19.Let P: This is a great website, Q: You should not come back here. Then ‘This is a great website and you should come back here.’ is best represented by?
a) ~P V ~Q
b) P  ~Q
c) P V Q
d) P ∧ Q

20.Let P: We should be honest., Q: We should be dedicated., R: We should be overconfident. Then ‘We should be honest or dedicated but not overconfident.’ is best represented by?
a) ~P V ~Q V R
b) P ∧ ~Q ∧ R
c) P V Q ∧ R
d) P V Q  ~R

21.Let P and Q be statements, then P<->Q is logically equivalent to __________
a) P<->~Q
b) ~P<->Q
c) ~P<->~Q
d) None of the mentioned

22.What is the negation of the statement A->(B v(or) C)?
a) A  ~B  ~C
b) A->B->C
c) ~A ∧ B v C
d) None of the mentioned

23.The compound statement A-> (A->B) is false, then the truth values of A, B are respectively _________
a) T, T
b) F, T
c) T, F
d) F, F

24.The statement which is logically equivalent to A (and) B is?
a) A->B
b) ~A ∧ ~ B
c) A ∧ ~B
d) ~(A->~B)

25.Let P: We give a nice overall squad performance, Q: We will win the match.

Then the symbolic form of “We will win the match if and only if we give a nice overall squad performance. “is?
a) P v Q
b) Q ∧ P
c) Q<->P
d) ~P v Q

26.Let P, Q, R be true, false true, respectively, which of the following is true?
a) P∧Q∧R
b) P∧~Q∧~R
c) Q->(PR)
d) P->(Q∧R)

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27.“Match will be played only if it is not a humid day.” The negation of this statement is?
a) Match will be played but it is a humid day
b) Match will be played or it is a humid day
c) All of the mentioned statement are correct
d) None of the mentioned

28.Consider the following statements.
A: Raju should exercise.
B: Raju is not a decent table tennis player.
C: Raju wants to play good table tennis.
The symbolic form of “Raju is not a decent table tennis player and if he wants to play good table tennis then he should exercise.” is?
a) A->B->C
b) B(C->A)
c) C->B∧A
d) B<->A∧C

29.The statement (~P<->Q)~Q is true when?
a) P: True Q: False
b) P: True Q: True
c) P: False Q: True
d) P: False Q: False

30.Let P, Q, R be true, false, false, respectively, which of the following is true?
a) P∧(Q∧~R)
b) (P->Q)∧~R
c) Q<->(PR)
d) P<->(QvR)

31.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

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32.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

33.The compound statement A v ~(A B).
a) True
b) False

Explanation: Applying De-Morgan’s law we get A v ~ A Ξ Tautology.

34.Which of the following is De-Morgan’s law?
a) P ∧ (Q v R) Ξ (P ∧ Q) v (P ∧ R)
b) ~(P  R) Ξ ~P v ~R, ~(P v R) Ξ ~P  ~R
c) P v ~P Ξ True, P ∧ ~P Ξ False
d) None of the mentioned

35.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)

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

Explanation: By identity A → B Ξ ~A V B.

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

38.Which of the following satisfies commutative law?
a) ∧
b) v
c) ↔
d) All of the mentioned

39.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

40.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

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

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

43.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

45.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

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

46.¬ (A q) (A q) is a ___________
a) Tautology
c) Contingency
d) None*/ of the mentioned

Explanation: ≡ (¬A ∧ ¬q) ∧ (A ∧ q)
≡ (¬A ∧ A) ∧ (¬q ∧ q)
≡ F ∧ F ≡ F.

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

Explanation: ≡ (A ∨ ¬A) ∨ (q ∨ T)
≡ T ∨ T ≡ T.

48.A ¬(A (A T)) is always __________
a) True
b) False

Explanation: ≡ A ∧ ¬ (A ∨ (A ∧ T))
≡ A ∧ ¬(A ∨ A)
≡ A ∧ ¬A ≡ F.

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

Explanation: ≡ (A ∨ F) ∨ (A ∨ T)
≡ A ∨ T ≡ T.

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

Explanation: ≡ A → (A ∨ q)
≡ ¬A ∨ (A ∨ q)
≡ (A ∨ ¬A) ∨ q
≡ T ∨ q ≡ T.

51.The contrapositive of p → q is the proposition of ____________
a) ¬p → ¬q
b) ¬q → ¬p
c) q → p
d) ¬q → p

52.The inverse of p → q is the proposition of ____________
a) ¬p → ¬q
b) ¬q → ¬p
c) q → p
d) ¬q → p

53.The converse of p → q is the proposition of _______________
a) ¬p → ¬q
b) ¬q → ¬p
c) q → p
d) ¬q → p

54.What is the contrapositive of the conditional statement? “The home team misses whenever it is drizzling?”
a) If it is drizzling, then home team misses
b) If the home team misses, then it is drizzling
c) If it is not drizzling, then the home team does not misses
d) If the home team wins, then it is not drizzling

Explanation: q whenever p contrapositive is ¬q → ¬p.

55.What is the converse of the conditional statement “If it ices today, I will play ice hockey tomorrow.”
a) “I will play ice hockey tomorrow only if it ices today.”
b) “If I do not play ice hockey tomorrow, then it will noz have iced today.”
c) “If it does not ice today, then I will not play ice hockey tomorrow.”
d) “I will not play ice hockey tomorrow only if it ices today.”

Explanation: If p, then q has converse q → p.

56.What are the contrapositive of the conditional statement “I come to class whenever there is going to be a test.”
a) “If I come to class, then there will be a test.”
b) “If I do not come to class, then there will not be a test.”
c) “If there is not going to be a test, then I don’t come to class.”
d) “If there is going to be a test, then I don’t come to class.”

Explanation: q whenever p, has contrapositive ¬q → ¬p..

57.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) “A positive integer is a composite if it has divisors other than 1 and itself.”
b) “If a positive integer has no divisors other than 1 and itself, then it is not composite.”
c) “If a positive integer is not composite, then it has no divisors other than 1 and itself.”
d) None of the mentioned

Explanation: p only if q has inverse ¬p → ¬q.

58.What are the converse of the conditional statement “When Raj stay up late, it is necessary that Raj sleep until noon.”
a) “If Raj stay up late, then Raj sleep until noon.”
b) “If Raj does not stay up late, then Raj does not sleep until noon.”
c) “If Raj does not sleep until noon, then Raj does not stay up late.”
d) “If Raj sleep until noon, then Raj stay up late.”

Explanation: Necessary condition for p is q has converse q → p.

60.What are the contrapositive of the conditional statement “Mediha will find a decent job when she labour hard.”?
a) “If Medha labour hard, then she will find a decent job.”
b) “If Medha will not find a decent job, then she not labour hard.”
c) “If Medha will find a decent job, then she labour hard.”
d) “If Medha not labour hard, then she will not find a decent job.”

Explanation: The statement q when p has its contrapositive as ¬q → ¬p.

61.What are the inverse of the conditional statement “If you make your notes, it will be a convenient in exams.”
a) “If you make notes, then it will be a convenient in exams.”
b) “If you do not make notes, then it will not be a convenient in exams.”
c) “If it will not be a convenient in exams, then you did not make your notes.”
d) “If it will be a convenient in exams, then you make your notes

Explanation: If p then q has inverse ¬p → ¬q.

62.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

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

Explanation: (p → q) ↔ (¬p ∨ q) is tautology.

64.p q is logically equivalent to ________
a) ¬q → ¬p
b) q → p
c) ¬p → ¬q
d) ¬p → q

Explanation: (p ∨ q) ↔ (¬p → q) is tautology.

65.¬ (p ↔ q) is logically equivalent to ________
a) q↔p
b) p↔¬q
c) ¬p↔¬q
d) ¬q↔¬p

Explanation: ¬(p↔q)↔(p↔¬q) is tautology.

66.p q is logically equivalent to ________
a) ¬ (p → ¬q)
b) (p → ¬q)
c) (¬p → ¬q)
d) (¬p → q)

Explanation: (p ∧ q) ↔ (¬(p → ¬q)) is tautology.

67.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

68.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)

Explanation: (p ↔ q) ↔ ((p → q) ∧ (q → p)) is tautology.

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

Explanation: ((p → q) ∧ (p → r)) ↔ (p → (q ∧ r)) is tautology.