JEE Mathematics Mathematical Reasoning MCQs Set 08

Answer: (d) p ∨ (q ∨ r)

Explanation:
1. S(~p, ~q, ~r) = p ∨ [~(~q ∧ ~r)].
2. By De Morgan, ~(~q ∧ ~r) = q ∨ r.
3. So the result is p ∨ (q ∨ r).

Teacher's Note:
1. Replace each variable by its negation and apply De Morgan's law.

Answer: (b) If a number is not a prime then it is not odd

Explanation:
1. The inverse of p → q is ~p → ~q.
2. Here p = 'a number is prime' and q = 'it is odd', so the inverse is: if a number is not a prime then it is not odd.

Teacher's Note:
1. Inverse negates both parts but keeps the order.

Answer: (d) Two plus two is four.

Explanation:
1. A statement is a sentence that is either true or false, but not both.
2. The first three are commands. 'Two plus two is four' is true, so it is a statement.

Teacher's Note:
1. Commands, questions and exclamations are not statements.

Answer: (a) T

Explanation:
1. '3 + 3 = 33 or 1 + 2 = 12' is false, since both parts are false.
2. Its negation, 'it is false that ...', is therefore true (T).

Teacher's Note:
1. Evaluate the inner 'or' first.

Answer: (a) Please do me a favour

Explanation:
1. A request cannot be called true or false, so it is not a statement.
2. The other three are definite statements (all true).

Teacher's Note:
1. Check whether each sentence has a definite truth value.

Answer: (b) Paris is not in France or London is not in England

Explanation:
1. ~(p ∧ q) = (~p) ∨ (~q) by De Morgan's law.
2. So the negation is: Paris is not in France or London is not in England.

Teacher's Note:
1. Negating an 'and' gives 'or' of the negations.

Answer: (a) \( (\sim q\wedge r)\to\sim p \)

Explanation:
1. The contrapositive of A → B is ~B → ~A.
2. Here B = (~q → ~r), and ~(~q → ~r) = ~q ∧ r.
3. So the contrapositive is (~q ∧ r) → ~p.

Teacher's Note:
1. Negate the whole consequent and swap.

Answer: (a) \( p\wedge(\sim q) \)

Explanation:
1. 'It is snowing' is p and 'it is not that I am cold' is ~q.
2. Joined by 'and' this is p ∧ (~q).

Teacher's Note:
1. Translate each clause and join with the connective.

Answer: (b) \( (p\to q)\equiv(\sim q\to\sim p) \)

Explanation:
1. (a) is false: ~p ∨ ~q = ~(p ∧ q).
2. (b) is the contrapositive law, always true.
3. (c) is false: ~(p → ~q) = p ∧ q. (d) is false: the right side is p ↔ q, not ~(p ∨ q).

Teacher's Note:
1. Test each option against a truth table or a standard law.

Answer: (c) \( (\sim p\vee q)\to\sim r \)

Explanation:
1. The inverse of A → B is ~A → ~B.
2. ~(p ∧ ~q) = ~p ∨ q, and ~r is the negated consequent.
3. So the inverse is (~p ∨ q) → ~r.

Teacher's Note:
1. Negate both antecedent and consequent.

Answer: (c) T, F

Explanation:
1. A conditional is false only when the antecedent is true and the consequent is false.
2. So p = T, and ~p ∨ q = F requires q = F (since ~p = F).

Teacher's Note:
1. A conditional is false only for T → F.

Answer: (a) Raju is not tall or he is intelligent

Explanation:
1. ~p is 'Raju is not tall' and q is 'Raju is intelligent'.
2. With ∨ meaning 'or', ~p ∨ q is: Raju is not tall or he is intelligent.

Teacher's Note:
1. ∨ means 'or'.

Answer: (b) It rains and I shall not go to school

Explanation:
1. ~(p → q) = p ∧ ~q.
2. So the negation is: it rains and I shall not go to school.

Teacher's Note:
1. A conditional fails only when the hypothesis holds and the conclusion fails.

Answer: (a) T, F, F

Explanation:
1. An implication is false only if p is true and q ∨ r is false.
2. q ∨ r is false only when both q and r are false.
3. So p = T, q = F, r = F.

Teacher's Note:
1. Use the single case where an implication is false.

Answer: (d) \( \sim(P\vee Q) \)

Explanation:
1. 'He is either intelligent or strong' is P ∨ Q.
2. 'It is not true that' negates the whole statement: ~(P ∨ Q).

Teacher's Note:
1. Negate the whole compound statement.