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JEE MainMathematics

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Mathematical Reasoning: Mathematics | JEE Main

Consider the following statements:

\mathrm{P} : \text{I have fever}

\mathrm{Q} : \text{I will not take medicine}

\mathrm{R} : \text{I will take rest.}

The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to:

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A

((\sim \mathrm{P}) \vee \sim \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \mathrm{R})

B

(\mathrm{P} \vee \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \mathrm{R})

C

((\sim \mathrm{P}) \vee \sim \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \sim \mathrm{R})

D

(\mathrm{P} \vee \sim \mathrm{Q}) \wedge (\mathrm{P} \vee \sim \mathrm{R})

Step-by-step solutionView

Correct answer

The given statement translates to $\mathrm{P} \rightarrow (\sim\mathrm{Q} \wedge \mathrm{R})$ , which is logically equivalent to $((\sim\mathrm{P}) \vee \sim\mathrm{Q}) \wedge ((\sim\mathrm{P}) \vee \mathrm{R})$ .

Option analysis

Why each option works or fails

A ·

((\sim \mathrm{P}) \vee \sim \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \mathrm{R})

None. This option correctly translates the given compound statement into propositional logic and applies standard equivalence laws. Recognize that 'I will take medicine' is $\sim\mathrm{Q}$ , making the statement $\mathrm{P} \rightarrow (\sim\mathrm{Q} \wedge \mathrm{R})$ , which expands to $(\sim\mathrm{P} \vee \sim\mathrm{Q}) \wedge (\sim\mathrm{P} \vee \mathrm{R})$ .

B ·

(\mathrm{P} \vee \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \mathrm{R})

The student failed to negate $\mathrm{P}$ in the first disjunct, mistakenly writing $\mathrm{P} \vee \mathrm{Q}$ instead of $\sim\mathrm{P} \vee \sim\mathrm{Q}$ , and ignored the negation on $\mathrm{Q}$ . Remember that the conditional law converts $A \rightarrow B$ to $\sim A \vee B$ ; the antecedent must always be negated.

C ·

((\sim \mathrm{P}) \vee \sim \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \sim \mathrm{R})

The student erroneously negated $\mathrm{R}$ when distributing the disjunction over the conjunction. Keep the truth values of the consequent components intact: 'I will take rest' is $\mathrm{R}$ , not $\sim\mathrm{R}$ .

D ·

(\mathrm{P} \vee \sim \mathrm{Q}) \wedge (\mathrm{P} \vee \sim \mathrm{R})

The student confused the conditional equivalence with its converse or contrapositive form, failing to negate the antecedent $\mathrm{P}$ and mistakenly negating $\mathrm{R}$ . Apply the rule $\mathrm{P} \rightarrow X \equiv \sim\mathrm{P} \vee X$ , rather than $\mathrm{P} \vee \sim X$ .

Reviewed route

Solution

StepWorking

01given

Given the primitive statements: $\mathrm{P} : \text{I have fever}$ $\mathrm{Q} : \text{I will not take medicine}$ $\mathrm{R} : \text{I will take rest.}$

02goal

Find a logically equivalent symbolic statement for: "If I have fever, then I will take medicine and I will take rest".

03approach

Translate the verbal statement into propositional logic: - "I have fever" is $\mathrm{P}$ . - "I will take medicine" is the negation of $\mathrm{Q}$ , i.e., $\sim \mathrm{Q}$ . - "I will take rest" is $\mathrm{R}$ . Thus, the compound statement is $\mathrm{P} \rightarrow (\sim \mathrm{Q} \wedge \mathrm{R})$ . Then rewrite the implication using $A \rightarrow B \equiv \sim A \vee B$ and apply the distributive law.

04execute

1. Express the implication as a disjunction: $\mathrm{P} \rightarrow (\sim \mathrm{Q} \wedge \mathrm{R}) \equiv (\sim \mathrm{P}) \vee (\sim \mathrm{Q} \wedge \mathrm{R})$ 2. Distribute $\vee$ over $\wedge$ : $(\sim \mathrm{P}) \vee (\sim \mathrm{Q} \wedge \mathrm{R}) \equiv ((\sim \mathrm{P}) \vee \sim \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \mathrm{R})$

✓verify

Check truth value when $\mathrm{P}$ is False (F): $\mathrm{P} \rightarrow \dots$ is vacuously True (T). In the derived expression, if $\mathrm{P} = \text{F}$ , then $\sim \mathrm{P} = \text{T}$ , making both $((\sim \mathrm{P}) \vee \sim \mathrm{Q})$ and $((\sim \mathrm{P}) \vee \mathrm{R})$ True, so their conjunction is True. This matches Option (0).

Hints that build this answer step by step

What is the symbolic representation of the statement: 'If I have fever, then I will take medicine and I will take rest'?
$\mathrm{P} \rightarrow (\sim\mathrm{Q} \wedge \mathrm{R})$
Using the conditional equivalence $A \rightarrow B \equiv \sim A \vee B$ , how is $\mathrm{P} \rightarrow (\sim\mathrm{Q} \wedge \mathrm{R})$ rewritten?
$\sim\mathrm{P} \vee (\sim\mathrm{Q} \wedge \mathrm{R})$
Applying the distributive law $A \vee (B \wedge C) \equiv (A \vee B) \wedge (A \vee C)$ to $\sim\mathrm{P} \vee (\sim\mathrm{Q} \wedge \mathrm{R})$ , what is the resulting expression?
$((\sim\mathrm{P}) \vee \sim\mathrm{Q}) \wedge ((\sim\mathrm{P}) \vee \mathrm{R})$

✓ Source and academic review

Question type: Single correct
Exam relevance: JEE Main · Mathematics
Concepts assessed: Mathematics
Academic status: Reviewed by official_key
Source: pyq
Editorial review: 9 September 2026

Quick checks

Students also ask

Why is 'I will take medicine' written as ~Q?

The stem defines Q as 'I will not take medicine'. Negating this gives ~Q, which means 'I will take medicine'.

Answer

The given statement translates to $\mathrm{P} \rightarrow (\sim\mathrm{Q} \wedge \mathrm{R})$ , which is logically equivalent to $((\sim\mathrm{P}) \vee \sim\mathrm{Q}) \wedge ((\sim\mathrm{P}) \vee \mathrm{R})$ .

Why each option works or fails

A · correct: $((\sim \mathrm{P}) \vee \sim \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \mathrm{R})$ - None. This option correctly translates the given compound statement into propositional logic and applies standard equivalence laws. Recognize that 'I will take medicine' is $\sim\mathrm{Q}$ , making the statement $\mathrm{P} \rightarrow (\sim\mathrm{Q} \wedge \mathrm{R})$ , which expands to $(\sim\mathrm{P} \vee \sim\mathrm{Q}) \wedge (\sim\mathrm{P} \vee \mathrm{R})$ .
B: $(\mathrm{P} \vee \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \mathrm{R})$ - The student failed to negate $\mathrm{P}$ in the first disjunct, mistakenly writing $\mathrm{P} \vee \mathrm{Q}$ instead of $\sim\mathrm{P} \vee \sim\mathrm{Q}$ , and ignored the negation on $\mathrm{Q}$ . Remember that the conditional law converts $A \rightarrow B$ to $\sim A \vee B$ ; the antecedent must always be negated.
C: $((\sim \mathrm{P}) \vee \sim \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \sim \mathrm{R})$ - The student erroneously negated $\mathrm{R}$ when distributing the disjunction over the conjunction. Keep the truth values of the consequent components intact: 'I will take rest' is $\mathrm{R}$ , not $\sim\mathrm{R}$ .
D: $(\mathrm{P} \vee \sim \mathrm{Q}) \wedge (\mathrm{P} \vee \sim \mathrm{R})$ - The student confused the conditional equivalence with its converse or contrapositive form, failing to negate the antecedent $\mathrm{P}$ and mistakenly negating $\mathrm{R}$ . Apply the rule $\mathrm{P} \rightarrow X \equiv \sim\mathrm{P} \vee X$ , rather than $\mathrm{P} \vee \sim X$ .

Step-by-step solution

given: Given the primitive statements: $\mathrm{P} : \text{I have fever}$ $\mathrm{Q} : \text{I will not take medicine}$ $\mathrm{R} : \text{I will take rest.}$
goal: Find a logically equivalent symbolic statement for: "If I have fever, then I will take medicine and I will take rest".
approach: Translate the verbal statement into propositional logic: - "I have fever" is $\mathrm{P}$ . - "I will take medicine" is the negation of $\mathrm{Q}$ , i.e., $\sim \mathrm{Q}$ . - "I will take rest" is $\mathrm{R}$ . Thus, the compound statement is $\mathrm{P} \rightarrow (\sim \mathrm{Q} \wedge \mathrm{R})$ . Then rewrite the implication using $A \rightarrow B \equiv \sim A \vee B$ and apply the distributive law.
execute: 1. Express the implication as a disjunction: $\mathrm{P} \rightarrow (\sim \mathrm{Q} \wedge \mathrm{R}) \equiv (\sim \mathrm{P}) \vee (\sim \mathrm{Q} \wedge \mathrm{R})$ 2. Distribute $\vee$ over $\wedge$ : $(\sim \mathrm{P}) \vee (\sim \mathrm{Q} \wedge \mathrm{R}) \equiv ((\sim \mathrm{P}) \vee \sim \mathrm{Q}) \wedge ((\sim \mathrm{P}) \vee \mathrm{R})$
verify: Check truth value when $\mathrm{P}$ is False (F): $\mathrm{P} \rightarrow \dots$ is vacuously True (T). In the derived expression, if $\mathrm{P} = \text{F}$ , then $\sim \mathrm{P} = \text{T}$ , making both $((\sim \mathrm{P}) \vee \sim \mathrm{Q})$ and $((\sim \mathrm{P}) \vee \mathrm{R})$ True, so their conjunction is True. This matches Option (0).