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Mathematical Logic (p-1) Maths (commerce) - Mathematical Logic (p-1)

Maharashtra Board - STD 12 Commerce / Arts - Maths (commerce) (MSBSHSE - English Medium). 192 questions in this chapter.

Question types

Mathematical Logic (p-1) question types

192 questions across 8 question groups — pick any mix to generate a Maths (commerce) paper with step-by-step answer keys.

192
Questions
8
Question groups
5
Question types
Sample Questions

Mathematical Logic (p-1) questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1MCQ1 Mark
Which of the following is always true?
  • A
    ~(p → q) ≡ ~q → ~p
  • B
    ~(p ∨ q) ≡ ~p ∨ ~q
  • ✓
    ~(p → q) ≡ p ∧ ~q
  • D
    ~(p ∧ q) ≡ ~p ∧ ~q

Answer: C.

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Q 3MCQ1 Mark
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
  • ✓
    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

Answer: B.

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Q 4MCQ1 Mark
If p : He is intelligent.
q : He is strong.
Then, symbolic form of statement: ‘It is wrong that, he is intelligent or strong’ is
  • A
    ~p ∨ ~p
  • B
    ~(p ∧ q)
  • ✓
    ~(p ∨ q)
  • D
    p ∨ ~q

Answer: C.

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Q 5MCQ1 Mark
Consider the following three statements
p : 2 is an even number.
q : 2 is a prime number.
r : Sum of two prime numbers is always even.
Then, the symbolic statement (p ∧ q) → ~r means:
  • A
    2 is an even and prime number and the sum of two prime numbers is always even.
  • B
    2 is an even and prime number and the sum of two prime numbers is not always even.
  • ✓
    If 2 is an even and prime number, then the sum of two prime numbers is not always even.
  • D
    If 2 is an even and prime number, then the sum of two prime numbers is also even.

Answer: C.

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Q 22Solve the Following Question.(2 Marks)2 Marks
If $A = {2, 3, 4, 5, 6, 7, 8}$, determine the truth value of each of the following statements:
(i) $∃ x \in A,$ such that $3x + 2 > 9$.
(ii) $\forall x \in A, x^2 < 18.$
(iii) $∃x \in A,$ such that $x + 3 < 11$.
(iv) $∀x \in A, x^2 + 2 \geq 5$.
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Q 23Solve the Following Question.(2 Marks)2 Marks
Consider the following statements:
(i) If D is a dog, then D is very good.
(ii) If D is very good, then D is a dog.
(iii) If D is not very good, then D is not a dog.
(iv) If D is not a dog, then D is not very good.
Identify the pairs of statements having the same meaning. Justify.
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Q 26Solve the Following Question.(3 Marks)3 Marks
Let p : Sachin win the match.
q : Sachin is a member of the Rajya Sabha.
r : Sachin is happy.
Write the verbal statement for each of the following:
(i) (p ∧ q) ∨ r
(ii) p → r
(iii) ~p ∨ q
(iv) p → (q ∨ r)
(v) p → q
(vi) (p ∧ q) ∧ ~r
(vii) ~(p ∨ q) ∧ r
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Q 27Solve the Following Question.(3 Marks)3 Marks
Rewrite the following statements without using the connective ‘If … then’:
(i) If a quadrilateral is a rhombus, then it is not a square.
(ii) If 10 – 3 = 7, then 10 × 3 ≠ 30.
(iii) If it rains, then the principal declares a holiday.
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Q 28Solve the Following Question.(3 Marks)3 Marks
State the dual of each of the following statements by applying the principle of duality:
(i) (p ∧ ~q) ∨ (~p ∧ q) ≡ (p ∨ q) ∧ ~(p ∧ q)
(ii) p ∨ (q ∨ r) ≡ ~[(p ∧ q) ∨ (r ∨ s)]
(iii) 2 is an even number or 9 is a perfect square.
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Q 31Solve the Following Question.(4 Marks)4 Marks
If p : Proof is lengthy.
q : It is interesting.
Express the following statements in symbolic form:
(i) Proof is lengthy and it is not interesting.
(ii) If the proof is lengthy, then it is interesting.
(iii) It is not true that the proof is lengthy but it is interesting.
(iv) It is interesting iff the proof is lengthy.
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Q 34Solve the Following Question.(4 Marks)4 Marks
Write the negation of each of the following statements:
(i) All the stars are shining if it is night.
(ii) $∀ n \in N, n + 1 > 0$.
(iii) $∃ n \in N$, such that $(n^2 + 2)$ is odd number.
(iv) Some continuous functions are differentiable.
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Q 35Solve the Following Question.(4 Marks)4 Marks
Write the dual statement of each of the following compound statements:
(i) 13 is prime number and India is a democratic country.
(ii) Karina is very good or everybody likes her.
(iii) Radha and Sushmita can not read Urdu.
(iv) A number is real number and the square of the number is non-negative.
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Q 36Solve the Following Question.(5 Marks)5 Marks
If p, q, r are statements with truth values T, T, F respectively, determine the truth values of the following:
(i) (p ∧ q) → ~p
(ii) p ↔ (q → ~p)
(iii) (p ∧ ~q) ∨ (~p ∧ q)
(iv) ~(p ∧ q) → ~(q ∧ p)
(v) ~[(p → q) ↔ (p ∧ ~q)]
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Q 37Solve the Following Question.(5 Marks)5 Marks
Assuming the following statements
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth values of the following:
(i) Stock prices are not high or stocks are rising.
(ii) Stock prices are high and stocks are rising if and only if stock prices are high.
(iii) If stock prices are high, then stocks are not rising.
(iv) It is false that stocks are rising and stock prices are high.
(v) Stock prices are high or stocks are not rising iff stocks are rising.
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Q 38Solve the Following Question.(5 Marks)5 Marks
Assuming the first statement as p and second as q, write the following statements in symbolic form:
(i) $x^3 + y^3 = (x + y)^3$^, iff $xy = 0.$
(ii) The drug is effective though it has side effects.
(iii) If a real number is not rational, then it must be irrational.
(iv) It is not true that Ram is tall and handsome.
(v) Even though it is not cloudy, it is still raining.
(vi) It is not true that intelligent persons are neither polite nor helpful.
(vii) If the question paper is not easy, then we shall not pass.
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Q 39Solve the Following Question.(5 Marks)5 Marks
Assuming the first statement as p and second as q, write the following statements in symbolic form:
(i) The Sun has set and Moon has risen.
(ii) Mona likes Mathematics and Physics.
(iii) 3 is a prime number if 3 is a perfect square number.
(iv) Kavita is brilliant and brave.
(v) If Kiran drives a car, then Sameer will walk.
(vi) The necessary condition for the existence of a tangent to the curve of the function is continuity.
(vii) To be brave is necessary and sufficient condition to climb Mount Everest.
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Q 40Solve the Following Question.(5 Marks)5 Marks
Which of the following sentences are statements? In case of a statement, write down the truth value:
(i) The Himalayas is the highest mountain range.
(ii) $(x – 2)(x – 3) = x^2 – 5x + 6$ for all $x \in R.$
(iii) What are the causes of rural unemployment?
(iv) $0! = 1.$
(v) The quadratic equation $ax^2 + bx + c = 0 (a \neq 0)$ always has two real roots.
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