InterviewSolution
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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Let p : A triangle is equilateral, q : A triangle is equiangular, then inverse of q → p is(A) If a triangle is not equilateral then it is not equiangular. (B) If a triangle is not equiangular then it is not equilateral. (C) If a triangle is equiangular then it is not equilateral. (D) If a triangle is equiangular then it is equilateral. |
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Answer» Correct option: (B) If a triangle is not equiangular then it is not equilateral. Inverse of q → p is ~q → ~p i.e., If a triangle is not equiangular then it is not equilateral. |
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| 2. |
Negation of (~p → q) is(A) ~ p ∨ ~q (B) ~p ∧~ q (C) p ∧ ~ q (D) ~ p ∨ q |
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Answer» Correct option: (B) ~p ∧~ q Since, p → q ≡ ~p ∨ q ∴ ~p → q ≡ p ∨ q ∴ ~(~p → q) ≡ ~(p ∨ q) ≡ ~p ∧~q |
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| 3. |
Write the negations of :All natural numbers are whole numbers. |
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Answer» The negation of the given statement is : 'Some natural numbers are not whole numbers.' |
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| 4. |
Write the duals of (p ∨ q) ∨ c. |
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Answer» The duals of the given statement patterns are : (p ∧ q) ∧ t |
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| 5. |
Write the negations of the x + 8 > 11 or y – 3 = 6 |
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Answer» Let p : x + 8 > 11, q : y — 3 = 6. Then, The symbolic form of the given statement is p ∨ q. Since ~(p ∨ q) ≡ ~p ∧ ~q, The negation of given statement is : ‘x + 8 > 11 and y – 3 ≠ 6’ OR ‘x + 8 </ 11 and y – 3 ≠ 6’ |
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| 6. |
Write the negations of the 11 < 15 and 25 > 20 |
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Answer» Let p: 11 < 15, q : 25 > 20. Then, The symbolic form of the given statement is p ∧ q. Since ~(p ∧ q) ≡ ~p ∨ ~q, The negation of given statement is : '11 ≮ 15 or 25 > 20.' OR '11 ≯ 15 or 25 </ 20.' |
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| 7. |
Write the negations of the :Qudrilateral is a square if and only if it is a rhombus. |
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Answer» Let p : Quadrilateral is a square. q : It is a rhombus. Then, The symbolic form of the given statement is p ↔ q. Since ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p), The negation of given statement is : 'Quadrilateral is a square but it is not a rhombus or quadrilateral is a rhombus but it is not a square.' |
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| 8. |
Write the negations of the :It is cold and raining. |
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Answer» Let p : It is cold. q : It is raining. Then, The symbolic form of the given statement is p ∧ q. Since ~(p ∧ q) ≡ ~p ∨ ~q, The negation of the given statement is : ‘It is not cold or not raining.’ |
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| 9. |
The statement, ‘If it is raining then I will go to college’ is equivalent to (A) If it is not raining then I will not go to college. (B) If I do not go to college, then it is not raining. (C) If I go to college then it is raining. (D) Going to college depends on my mood. |
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Answer» Correct option: (B) If I do not go to college, then it is not raining. r: It is raining, c: I will go to college. The given statement is r → c ≡ ~c → ~r |
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| 10. |
p → Ram is richq → Ram is successfulr → Ram is talentedWrite the symbolic form of the given statement.Ram is neither rich nor successful and he is not talented(A) ~p ∧ ~q ∨ ~r (B) ~p ∨ ~q ∧ ~r (C) ~p ∨ ~q ∨ ~r (D) ~p ∧ ~q ∧ ~r |
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Answer» Correct option: (D) ~p ∧ ~q ∧ ~r ~ p : Ram is not rich ~ q : Ram is not successful ~ r : Ram is not talented ∴ The symbolic form of the given statement is ~p ∧ ~q ∧ ~r. |
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| 11. |
p : There are clouds in the sky and q : it is not raining. The symbolic form is (A) p → q (B) p → ~q (C) p ∧ ~q (D) ~p ∧ q |
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Answer» Correct option: (C) p ∧ ~q p: There are clouds in the sky, ~q: It is not raining, ‘and’ is expressed by ‘∧’ symbol. ∴ p ∧ ~q |
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| 12. |
If first part of the sentence is p and the second is q, then the symbolic form of the statement ‘It is not true that Physics is not interesting or difficult’ is(A) ~(~p ∧ q) (B) (~p ∨ q) (C) (~p ∨ ~q) (D) ~(~p ∨ q) |
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Answer» Correct option: (D) ~(~p ∨ q) p: Physics is interesting. q: Physics is difficult. ∴ Symbolic form: ~ (~p ∨ q) |
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| 13. |
The symbolic form of the statement ‘It is not true that intelligent persons are neither polite nor helpful’ is (A) ~(p ∨ q) (B) ~(~p ∧ ~q) (C) ~(~p ∨ ~q) (D) ~(p ∧ q) |
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Answer» Correct option: (B) ~(~p ∧ ~q) p: Intelligent persons are polite. q: Intelligent persons are helpful. ∴ Symbolic form: ~ (~p ∧ ~q) |
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| 14. |
Let p be the proposition : Mathematics is interesting and let q be the proposition : Mathematics is difficult, then the symbol p ∧ q means(A) Mathematics is interesting implies that Mathematics is difficult. (B) Mathematics is interesting implies and is implied by Mathematics is difficult. (C) Mathematics is interesting and Mathematics is difficult. (D) Mathematics is interesting or Mathematics is difficult. |
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Answer» Correct option: (C) Mathematics is interesting and Mathematics is difficult. The symbol p ∧ q means Mathematics is interesting and Mathematics is difficult. |
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| 15. |
Let S be a non-empty subset of R. Consider the following statement:p : There is a rational number x ∈ S such that x > 0.Which of the following statements is the negation of the statement p?(A) There is a rational number x ∈ S such that x ≤ 0 (B) There is no rational number x ∈ S such that x ≤ 0 (C) Every rational number x ∈ S satisfies x ≤ 0 (D) x ∈ S and x ≤ 0 → x is not rational |
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Answer» Correct option: (C) Every rational number x ∈ S satisfies x ≤ 0 Given statement is ∃ x ∈ S, such that x > 0 ∴ ~ (∃ x ∈ S, such that x > 0) ≡ ∀ x ∈ S, x ≤ 0 i.e., Every rational number x ∈ S satisfies x ≤ 0. |
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| 16. |
Write the truth values of the following :(i) 4 is odd or 1 is prime.(ii) 64 is a perfect square and 46 is a prime number.(iii) 5 is a prime number and 7 divides 94.(iv) It is not true that 5 – 3i is a real number.(v) If 3 × 5 = 8, then 3 + 5 = 15.(vi) Milk is white if and only if sky is blue.(vii) 24 is a composite number or 17 is a prime number. |
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Answer» (i) Let p : 4 is odd. q : 1 is prime. Then The symbolic form of the given statement is p∨q. The truth values of both p and q are F. ∴ the truth value of p v q is F. … [F ∨ F = F] (ii) Let p : 64 is a perfect square. q : 46 is a prime number. Then, The symbolic form of the given statement is p∧q. The truth values of p and q are T and F respectively. ∴ the truth value of p ∧ q is F. … [T ∧ F ≡ F] (iii) Let p : 5 is a prime number. q : 7 divides 94. Then the symbolic form of the given statement is p∧q. The truth values of p and q are T and F respectively. ∴ the truth value of p ∧ q is F. … [T ∧ F ≡ F] (iv) Let p : 5 – 3i is a real number. Then, The symbolic form of the given statement is ~ p. The truth values of p is F. ∴ the truth values of ~ p is T. … [~ F ≡ T] (v) Let p : 3 × 5 = 8. q : 3 + 5 = 15. Then, The symbolic form of the given statement is p → q. The truth values of both p and q are F. ∴ the truth value of p → q is T. … [F → F ≡ T] (vi) Let p : Milk is white. q : Sky is blue. Then, The symbolic form of the given statement is p ↔ q. The truth values of both p and q are T. ∴ the truth value of p ↔ q is T. … [T ↔ T ≡ T] (vii) Let p : 24 is a composite number. q : 17 is a prime number. Then, The symbolic form of the given statement is p ∨ q. The truth values of both p and q are T. ∴ the truth value of p ∨ q is T. … [T ∨ T ≡ T] |
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| 17. |
The negation of the proposition “If 2 is prime, then 3 is odd” is(A) If 2 is not prime, then 3 is not odd. (B) 2 is prime and 3 is not odd. (C) 2 is not prime and 3 is odd. (D) If 2 is not prime then 3 is odd. |
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Answer» Correct option: (B) 2 is prime and 3 is not odd. Let p : 2 is prime, q : 3 is odd ∴ Symbolic form p → q ∴ ~(p → q) ≡ p ∧ ~q i.e., 2 is prime and 3 is not odd. |
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| 18. |
Which of the following is a statement? (A) Stand up!(B) Will you help me?(C) Do you like social studies?(D) 27 is a perfect cube. |
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Answer» Correct answer: (D) 27 is a perfect cube. |
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| 19. |
Which of the following is an open statement? (A) x is a natural number.(B) Give me a glass of water.(C) Wish you best of luck.(D) Good morning to all. |
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Answer» Correct answer: (A) x is a natural number |
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| 20. |
Negate the following. (i) 4 is an even integer or 7 is a prime number. (ii) He likes to run and he does not like to sit. (iii) He likes Mathematics and he does not like Logic. (iv) If 6 is a divisor of 120 then 486 is not dividiable by 6. (v) If 2 triangles are similar then their ares are equal. (vi) It is cold or it is raining, |
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Answer» (i) Let p: 4 is even integer, q: 7 is a prime number given (p v q) ~(p ∨ q) = ~ p ∧ ~q 4 is not an even integer & 7 is not a prime number (ii) Let p: He likes to run, q: He likes to sit Given is ~ p ∧ ~q ∴ ~ (p ∧~ ) = ~p ∨ ~(~q) ≡ ~p ∨ q He does not like to run or he likes to sit (iii) Let p: He likes mathematices, q: He like logic Given (p ∧ ~q) ~ (p ∧ ~q) ≡ ~p ∨ q He does not like mathematics or he likes logic (iv) Let p: 6 is a divisor or 120 q: 486 is divisible by 6 Given (p → ~ q) ~(p → ~ q) = p ∧ ~(~q) (v) Let p: 2 triangles are similar. q: Areas are equal Given (p → q) ∴ ~(p → q) = (p ∧ ~q) 2 triangles are similar & areas are not equal. (vi) p: It is cold q: It is raining Given p ∨ a ~(p ∨ q) =~p∧~q It is not cold & it is not raining |
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| 21. |
Which of the following is not a statement?(A) Please do me a favour.(B) 2 is an even integer.(C) 2 + 1 = 3.(D) The number 17 is prime. |
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Answer» Correct answer: (A) Please do me a favour. |
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| 22. |
Let p : 7 is not greater than 4 and q : Paris is in France be two statements. Then, ~(p ∨ q) is the statement(A) 7 is greater than 4 or Paris is not in France. (B) 7 is not greater than 4 and Paris is not in France. (C) 7 is not greater than 4 and Paris is in France. (D) 7 is greater than 4 and Paris is not in France. |
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Answer» Correct option: (D) 7 is greater than 4 and Paris is not in France. ~(p ∨ q) ≡ (~p) ∧ (~q) i.e., 7 is greater than 4 and Paris is not in France. |
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| 23. |
Negate (i) p → (q ∧ r) (ii) q ∨ [~(p ∧ r)] (iii) (p → q)∧( q → p/q) (iv) p→ (q ∧ ~r) |
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Answer» (i) ~ [p → (p ∧ r)] = p ∧~(q ∧ r) ≡ p ∧ (~q ∨ ~r) (ii) ~ [q ∨ (~ (p ∧ r)] ≡ ~q ∧ ~[~ (p ∧ r) ≡ ~q ∧ (p ∧ r) (iii) ~[(p →q) ∧ (q → p)] ≡ ~ (p →q) ∨ ~(q → p) ≡ (p ∧ ~q) ∨ (q ∧ ~p) (iv) ~[p → (q^~r)] = p ∧ ~(q ∧ ~r) = p ∧ [(-q ∨ ~(~r)] = p ∧(~q ∨ r) |
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| 24. |
Negate the following: (1) If an integer is greater than 3 and less than 5 then it is a multiple of 5. (ii) If ‘x’ is divisible by ‘y’ then it is divisible by ‘a’ and ‘b. (iii) Weather is fine and my friends are not coming or we do not go to a movie. (iv) If a triangle is equilateral then it’s sides are equal and angles are equal. (v) 14 is a divisor of 48 and 28 is not divisible by 82. |
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Answer» (i) Let p: An integer is greater than 3 q: An integer is less than 5 r: An integer is multiple of 5 Given[(p ∧ q) → r] ∴ ~ ((p ∧ q) → r] = (p ∧ q) ∧~r An integer greater than 3 and less than 5 and not divisible by b (ii) Let P:x is divisible by y q: x is divisible by a r: x is divisible by b Given [p → (q ∧ r)] ∴ ~P (q ∧ r)] = p ∧~(q ∧ ~r) = p ∧( ~ ( q ∨ ~r) x is divisible by y and x is not divisible by a or not a multiple of 5 (iii) Let p: Weather is fine q: Friends are coming r: we go to a movie Given[p∧~q ∨~r)] ~(p ∧ (~q ∨ ~r)) ≡ ~p ∨ ~(~ q ∨ ~r) : ≡ ~p ∨ (q ∧ r) (iv) Let p = a triangle is equilateral q: Sides are equal r: angles are equal Given p → (q ∧ r) ∴ ~(p → (~q ∨ ~r)) ≡ ~p ∧ ~q ∨ ~r) A triangle is equilateral and sides are not. equal or angles are not equal. Let p: 14 is a divisor of 48 q: 28 is divisible by 82 Given p ∧ ~q ∴ ~(p ∧ ~q) ≡ ~p ∨ q 14 is not a divisor of 48 or 28 is divisible by 82 |
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| 25. |
Show that (p → q) + (~q→~p) is a Tautology. |
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Answer» (p → q) + (~q→~p)
It is a tautology |
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| 26. |
The statement (p \(\land\) q) \(\land\) (~ p \(\lor\) ~ q) is(A) a contradiction.(B) a tautology.(C) neither a contradiction nor a tautology.(D) equivalent to p \(\lor\) q. |
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Answer» Correct answer: (A) a contradiction. |
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| 27. |
Check whether the following propositions is a Tautology or a contradiction. (i) (p ∧ ~q) → (p ∧ q) (ii) [~p ∧ (p ∨ q)] → q (iii) (p →q) ↔ (~p →~q) (iv) [~(p →~q)] ∨ (~ p ↔ q) (v) (~p ∨ q) ↔ (p ∨~9) |
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Answer» (i) (p ∧~q) → (p∧q)
It is neither Tautology nor contradiction (ii) [~p ∧ (p ∨ q)] → q
F From last column we conclude it is a tautology (iii) (p →q) ↔ (~p →~q)
F It is neither a tautology nor contradiction (iv) [~(p →~q)] ∨ (~ p ↔ q)
It is neither tautology nor contradiction (v) (~p ∨ q) ↔ (p ∨~q)
It is neither tautology nor contradiction |
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| 28. |
Determine whether the following propositions is a Tautology or a contradiction or neither. (i) (p ∧ q) ∧ ~p (ii) [~p ∧ (p ∨ q)] (iii) (p ∧ q) → (p ∨ q) (iv) (p ∧ q) →p (v) ~ p ∧ ~q |
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Answer» (i) (p ∧ q) ∧~p
From last column we conclude that it is a contradiction (ii) [~p ∧ (p ∨ q)]
From last column we conclude it is neither tautology nor a contradiction (iii) (p ∧ q) → (p ∨ q)
From last column we conclude it is a tautology (iv) (p ∧ q) → p
From last column we conclude that it is a tautology (v) ~ p ∧ ~q
It is neither tautology nor contradiction |
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| 29. |
If p and q are simple propositions, then p \(\land\) q is true when(A) p is true and q is false.(B) p is false and q is true.(C) p is true and q is true.(D) p is false q is false. |
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Answer» Correct answer: (C) p is true and q is true. |
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| 30. |
The connective in the statement “Earth revolves around the Sun and Moon is a satellite of earth”, is (A) or (B) Earth (C) Sun (D) and |
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Answer» Correct option: (D) and |
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| 31. |
If p \(\land\) q = F, p \(\rightarrow\) q = F, then the truth value of p and q is :(A) T, T(B) T, F(C) F, T(D) F, F |
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Answer» Correct answer: (B) T, F |
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| 32. |
Which of the following is a statement in logic? (A) Go away (B) How beautiful! (C) x > 5 (D) 2 = 3 |
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Answer» Correct option: (D) 2 = 3 Even though 2 = 3 is false, it is a statement in logic with truth value F. |
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| 33. |
Which of the following is a statement in logic?(A) What a wonderful day! (B) Shut up! (C) What are you doing? (D) Bombay is the capital of India. |
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Answer» Correct option: (D) Bombay is the capital of India. ‘Bombay is the capital of India’ is a statement. The other options are exclamatory and interrogative sentences. |
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| 34. |
Which of the following is a statement? (A) Open the door. (B) Do your homework. (C) Switch on the fan. (D) Two plus two is four. |
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Answer» Correct option: (D) Two plus two is four. ‘Two plus two is four’ is a statement. The other options are imperative sentences. |
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| 35. |
State which of the following sentences are statements. Justify your answer.In case of statement, write down the truth value :(i) 5 + 4 = 13.(ii) x – 3 = 14.(iii) Close the door.(iv) Zero is a complex number.(v) Please get me breakfast.(vi) Congruent triangles are also similar.(vii) x2 = x.(viii) A quadratic equation cannot have more than two roots.(ix) Do you like Mathematics ?(x) The sun sets in the west.(xi) All real numbers are whole numbers.(xii) Can you speak in Marathi ?(xiii) x2 – 6x – 7 = 0, when x = 7.(xiv) The sum of cuberoots of unity is zero.(xv) It rains heavily. |
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Answer» (i) It is a statement which is false, hence its truth value is ‘F’. (ii) It is an open sentence, hence it is not a statement. (iii) It is an imperative sentence, hence it is not a statement. (iv) It is a statement which is true, hence its truth value is ‘T’. (v) It is an imperative sentence, hence it is not a statement. (vi) It is a statement which is true, hence its truth value is ‘T’. (vii) It is an open sentence, hence it is not a statement, (viii) It is a statement which is true, hence its truth value is ‘T’. (ix) It is an interrogative sentence, hence it is not a statement. (x) It is a statement which is true, hence its truth value is ‘T’. (xi) It is a statement which is false, hence its truth value is ‘F’. (xii) It is an interrogative sentence, hence it is not a statement. (xiii) It is a statement which is true, hence its truth value is ‘T’. (xiv) It is a statement which is true, hence its truth value is ‘T’. (xv) It is an open sentence, hence it is not a statement. |
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| 36. |
The contrapositive of (p \(\lor\) q) \(\rightarrow\) r is (A) ~r \(\rightarrow\) ~p \(\land\) ~q(B) ~r \(\rightarrow\)(p \(\lor\) q)(C) r \(\rightarrow\)(p \(\lor\) q)(D) p \(\rightarrow\) (q \(\lor\) r) |
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Answer» Correct answer: (A) ~r \(\rightarrow\) ~p \(\land\) ~q |
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| 37. |
The logically equivalent statement of ~p \(\lor\) ~q is(A) ~p \(\land\) ~q(B) ~(p \(\land\) q)(C) ~(p \(\lor\) q)(D) p \(\land\) q |
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Answer» Correct answer: (B) ~(p \(\land\) q) |
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| 38. |
The logically equivalent statement of p \(\rightarrow\) q is(A) ~p \(\lor\) q(B) q \(\rightarrow\) ~p(C) ~q \(\lor\) p(D) ~q \(\lor\) ~p |
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Answer» Correct answer: (A) ~p \(\lor\) q |
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| 39. |
(p ∧ q) → r is logically equivalent to ________.(a) p → (q → r)(b) (p ∧ q) → ~r (c) (~p ∨ ~q) → ~r (d) (p ∨ q) → r |
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Answer» Option : (a) p → (q → r) [Hint: Use truth table.] |
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| 40. |
Write the truth values of the following statements :\(\forall\) n ∈ N, n2 + n is even number while n2 – n is an odd number. |
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Answer» Let p : \(\forall\) n ∈ N, n2 + n is an even number. q : \(\forall\) n ∈ N, n – n is an odd number. Then, The symbolic form of the given statement is p ∧ q. The truth values of p and q are T and F respectively. ∴ The truth value of p ∧ q is F. … [T ∧ F ≡ F]. |
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| 41. |
The converse of ‘If x is zero then we cannot divide by x’ is (A) If we cannot divide by x then x is zero. (B) If we divide by x then x is non-zero. (C) If x is non-zero then we can divide by x. (D) If we cannot divide by x then x is non-zero. |
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Answer» Correct option: (A) If we cannot divide by x then x is zero. Converse of p → q is q → p. |
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| 42. |
If p : Prakash passes the exam, q : Papa will give him a bicycle. Then the statement ‘Prakash passing the exam, implies that his papa will give him a bicycle’ can be symbolically written as (A) p → q (B) p ↔ q (C) p ∧ q (D) p ∨ q |
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Answer» Correct option: (A) p → q “Implies” is expressed as ‘→’. ∴ symbolic form is p → q |
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| 43. |
The converse of the statement “If x > y, then x + a > y + a”, is (A) If x < y, then x + a < y + a (B) If x + a > y + a, then x > y (C) If x < y, then x + a > y + a (D) If x > y, then x + a < y + a |
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Answer» Correct option: (B) If x + a > y + a, then x > y Let p : x > y q : x + a > y + a Converse of p → q is q → p i.e., If x + a > y + a, then x > y |
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| 44. |
For the statements p and q ‘p → q’ is read as ‘if p then q’. Here, the statement q is called (A) antecedent. (B) consequent. (C) logical connective. (D) prime component. |
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Answer» Correct option: (B) consequent. |
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| 45. |
If p: Rohit is tall, q: Rohit is handsome, then the statement ‘Rohit is tall or he is short and handsome’ can be written symbolically as (A) p ∨ (~p ∧ q) (B) p ∧ (~p ∨ q) (C) p ∨ (p ∧ ~q) (D) ~p ∧ (~p ∧ ~q) |
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Answer» Correct option: (A) p ∨ (~p ∧ q) ~p: Rohit is short, ‘or’ is expressed by ‘∨’ symbol, ‘and’ is expressed by ‘∧’ symbol. |
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| 46. |
The inverse of the statement “If you access the internet, then you have to pay the charges”, is (A) If you do not access the internet, then you do not have to pay the charges. (B) If you pay the charges, then you accessed the internet. (C) If you do not pay the charges, then you do not access the internet. (D) You have to pay the charges if and only if you access the internet. |
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Answer» Correct option: (A) If you do not access the internet, then you do not have to pay the charges. Let p: You access the internet q: You have to pay the charges Given statement is written symbolically as, p → q Inverse of p → q is ~p → ~q i.e. If you do not access the internet then you do not have to pay the charges. |
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| 47. |
A compound statement p and q is true only when (A) p is true. (B) q is true. (C) both p and q are true. (D) none of p and q is true. |
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Answer» Correct option: (C) both p and q are true |
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| 48. |
Assuming the first part of the statement as p, second as q and the third as r, the statement ‘Candidates are present, and voters are ready to vote but no ballot papers’ in symbolic form is (A) (p ∨ q) ∧ ~r (B) (p ∧ ~q) ∧ r (C) (~p ∧ q) ∧ ~r (D) (p ∧ q) ∧ ~r |
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Answer» Correct option: (D) (p ∧ q) ∧ ~r p: Candidates are present, q: Voters are ready to vote r: Ballot papers ⇒ ~r : no Ballot papers. ‘and’ and ‘but’ are represented by ‘∧’ symbol. |
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| 49. |
The contrapositive of the statement: “If a child concentrates then he learns” is (A) If a child does not concentrate he does not learn. (B) If a child does not learn then he does not concentrate. (C) If a child practises then he learns. (D) If a child concentrates, he does not forget. |
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Answer» Correct option: (B) If a child does not learn then he does not concentrate. Contrapositive of p → q is ~q → ~p. |
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| 50. |
Write verbally ~p ∨ q where p: She is beautiful; q: She is clever (A) She is beautiful but not clever (B) She is not beautiful or she is clever (C) She is not beautiful or she is not clever (D) She is beautiful and clever. |
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Answer» Correct option: (B) She is not beautiful or she is clever ~p: She is not beautiful, ‘∨’ indicates ‘or’. |
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