JEE Mathematics Mathematical Reasoning MCQs Set 13

Answer: (d) \( p\vee(q\vee r) \)

Explanation:
1. Replace p, q, r by their negations: S = p ∨ ~(~q ∧ ~r).
2. By De Morgan, ~(~q ∧ ~r) = q ∨ r.
3. So S(~p,~q,~r) = p ∨ (q ∨ r).

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

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

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (d) Two plus two is four.

Explanation:
1. A statement (proposition) is a sentence that is either true or false, but not both.
2. The first three are commands, which have no truth value.
3. 'Two plus two is four' is true, so it is a statement.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (a) T

Explanation:
1. 3 + 3 = 33 is false and 1 + 2 = 12 is false, so the 'or' statement is false.
2. The negation of a false statement is true, so the given statement is true.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

Explanation:
1. A contradiction is false for every truth assignment.
2. In (c), q and ~q cannot both be true, so the statement is always false.
3. The other options are true for some assignments (for instance (b) is a tautology).

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (a) Please do me a favour

Explanation:
1. A request has no truth value, so it is not a statement.
2. The other three are true statements.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

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

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

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

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

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

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (d) none of these

Explanation:
1. All three given sentences say the same thing: that New Delhi is not a city.
2. So each is a negation of p, which means 'none of these' is the correct choice for 'not a negation'.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (c) Mathematics is interesting

Explanation:
1. A proposition must be definitely true or false.
2. 'Mathematics is interesting' is a matter of opinion, so it has no definite truth value.
3. The others are definitely true or false.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

Explanation:
1. (a) is false: by De Morgan ~p ∨ ~q = ~(p ∧ q).
2. (b) is the contrapositive law, which always holds.
3. (c) is false since ~(p implies ~q) = p ∧ q. (d) is false since the right side is p iff q.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (b) \( p\to(p\vee q) \)

Explanation:
1. p → (q → p) = ~p ∨ (~q ∨ p), which is always true, a tautology.
2. p → (p ∨ q) = ~p ∨ p ∨ q is also always true.
3. The other options are false when p is true and q is false.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

Explanation:
1. The inverse of A implies B is ~A implies ~B.
2. ~(p ∧ ~q) = ~p ∨ q by De Morgan.
3. So the inverse is (~p ∨ q) implies ~r.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (c) T, F

Explanation:
1. An implication is false only when the antecedent is true and the consequent false.
2. So p is T and (~p ∨ q) is F, which needs ~p = F (true since p = T) and q = F.
3. Thus p = T and q = F.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

Explanation:
1. ~p is 'Raju is not tall'.
2. The symbol ∨ means 'or'.
3. So ~p ∨ q reads: Raju is not tall or he is intelligent.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

Explanation:
1. The negation of p implies q is p ∧ ~q.
2. So: it rains and I shall not go to school.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (a) T, F, F

Explanation:
1. An implication is false only when the antecedent is true and the consequent is false.
2. So p = T and q ∨ r = F, which means q = F and r = F.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q 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).
3. Option (b) is equivalent by De Morgan but the direct translation of the sentence is ~(P ∨ Q).

Teacher's Note:
1. Translate the sentence literally first, then simplify if needed.
2. Several options can be logically equivalent; choose the direct symbolic form.

Answer: (b) If Ashok does not work hard then he gets good grade

Explanation:
1. ~p means 'Ashok does not work hard' and q means 'Ashok gets good grade'.
2. The arrow means 'if ... then', so the statement reads: If Ashok does not work hard then he gets good grade.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (d) T, F

Explanation:
1. The implication is false only when p = T and (~p ∨ q) = F.
2. With p = T, ~p is F, so ~p ∨ q = F requires q = F.
3. So p = T and q = F.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (c) \( (p\wedge q)\wedge(p\wedge\sim r) \)

Explanation:
1. The negation of A implies B is A ∧ ~B.
2. ~B = ~(~p ∨ r) = p ∧ ~r.
3. So the negation is (p ∧ q) ∧ (p ∧ ~r).

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (c) \( \sim[p\vee(\sim q)]\equiv(\sim p)\wedge q \)

Explanation:
1. By De Morgan, ~[p ∨ ~q] = ~p ∧ ~(~q) = ~p ∧ q, so (c) is true and (d) is false.
2. (a) is false since ~[p ∧ ~q] = ~p ∨ q.
3. (b) is false since the left side is (~p ∧ ~q) ∨ r.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (b) a contradiction

Explanation:
1. p ∧ ~p is always false, so the whole conjunction is always false.
2. A statement that is always false is a contradiction.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (d) \( p\Rightarrow q \) is true

Explanation:
1. p ∧ q = F ∧ T = F, and q ∧ p = F.
2. p ∨ ~q = F ∨ F = F.
3. p ⇒ q with a false antecedent is true.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (c) contradiction

Explanation:
1. p ⇒ ~p is equivalent to ~p, and ~p ⇒ p is equivalent to p.
2. The conjunction is p ∧ ~p, which is always false.
3. So it is a contradiction.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (c) a circle is not an ellipse

Explanation:
1. The negation of a statement says the opposite.
2. So: a circle is not an ellipse.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (d) \( ((p\wedge q)\to q)\to p \) is a tautology

Explanation:
1. (p ∧ q) → q is always true, so the statement in (d) reduces to T → p, which equals p.
2. p is not true for every assignment, so (d) is not a tautology.
3. (a), (b), (c) are standard true statements.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (a) TTFF

Explanation:
1. TT: (T) iff (F ∨ T = T) gives T.
2. TF: (F) iff (F ∨ F = F) gives T.
3. FT: (F) iff (T ∨ T = T) gives F. FF: (F) iff (T) gives F.
4. The column is T, T, F, F.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

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

Explanation:
1. If p ∧ q is true then p ∨ q is true, so ~(p ∨ q) is false.
2. If p ∧ q is false the conjunction is already false.
3. So (a) is false always, a contradiction.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (b) a tumbler is half empty if and only if it is half full

Explanation:
1. 'p if and only if q' is the biconditional p ⟺ q.
2. Reading it in words: a tumbler is half empty if and only if it is half full.

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.

Answer: (c) \( \sim r\Rightarrow\sim p\wedge\sim q \)

Explanation:
1. The contrapositive of A implies B is ~B implies ~A.
2. ~(p ∨ q) = ~p ∧ ~q by De Morgan.
3. So: ~r implies (~p ∧ ~q).

Teacher's Note:
1. Build the truth table or use the standard laws (De Morgan, contrapositive) to compare statements.
2. Remember: p implies q is false only when p is true and q is false.