MATHEMATICAL REASONING
Statement logical operations and, or, implies, implied by, if and only if, understanding of tautology, contradiction, converse and contrapositive.
- The statement among the following that is a tautology is : (1) A ∧ (A ∧ B) (2) A ∧(A ∧ B) (3) B → [A ∧(A → B)] (4) [A ∧ (A → B)] → B Correct Answer is (4) [A ∧ (A → B)] → B 2
- The negation of the statement ∼p∧(p∨q) is : A ) p∨∼q B) ∼p∨q C) ∼ p ∧ q D) p ∧ ∼ q Option (A) is correct A ) p∨∼q 2
- The statement A → (B → A) is equivalent to : (1) A → (A ∧ B) (2) A →(A → B) (3) A → Option 4) is correct (4) A → (A V B) 2
- The contrapositive of the statement "If you will work, you will earn money" is : (1) You will earn money, Correct answer is (3) If you will not earn money, you will not work Contrapositive of p → q is ~q → ~p p : you will work q : you will earn m... 1
- Let F1(A,B,C) = (A∧~B) ∨ [~C ∧ (A ∨ B)] ∨~A and F2(A, B) = (A ∨ B) ∨ (B →~A) be two logical expressions. Correct option is (3) (3) F₁ is not tautology but F₂ is a tautology F₁ : (A ∧~B) ∨[~C ∧ (A ∨ B)] ∨~A F₂ : (A ∨ B) ∨ (B →~A) F₁ : (A ∧~B) ∨~A... 1
- Which of the following Boolean expression is a tautology? (1) (p ∧ q) ∨ (p ∨ q) Correct Answer is 4) (p ∧ q) → (p → q) 1
- If the Boolean expression (p ⇒ q) ⇔ (q * ( ∼ q) is a tautology, then the boolean expression p * ( ∼ q) is equivalent to : Answer: (a) (p → q) ⇔ (q * ~ P) p q p → q q * ~ P ~ q ~ q ∧ p ~ (~ q ∧ p) T T T T F F T T F F F T T F F T T T F F T F F T T T F T ∴ ~ (~ q ∧... 1
- If the Boolean expression ( p ∧ q ) ⊙ ( p ⊗ q ) is a tautology, then ⊙ and ⊗ are respectively given by : Option 1 is Correct ( →, →) ( p ∧ q ) → ( p → q ) ( p ∧ q ) → ( ∼ p ∨ q ) ( ∼ p ∨ ∼ q ) ∨ ( ∼ p ∨ q ) ∼ p ∨ ( ∼ q ∨ q ) ⇒ Tautology ⇒ ⊙ ⇒→ ⊗... 1
- If P and Q are two statements, then which of the following compound statement is a tautology ? Answer: (b) (P ⇒ Q) ^ ~ Q = (~ P ∨ Q) ^ ~ Q ≡ (~ P ∧ ~ Q) ∨ (Q ∧ ~ Q) = (~ P ∨ ~ Q) ∨ (Q ^ ~ Q) = ~ (P ∨ Q) Now, (1) ~ (P v Q) ⇒ P = (P v Q)... 1