Why is (~ p vee q) vee (~ p vee r) the same as ~ p vee (q vee r)?

Disjunction vee is associative and commutative, so the expression is simply ~ p vee ~ p vee q vee r. By idempotence, ~ p vee ~ p ≡ ~ p, yielding ~ p vee (q vee r).

JEE MainMathematics

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+4 marks−1 if incorrectSingle correctPrevious-year question

Logical Equivalence of Disjunctive and Conditional Compound Statements

Let p, q, r be three logical statements. Consider the compound statements S_1:((~ p) vee q) vee((~ p) vee r) and S_2: p →(q vee r) Then, which of the following is NOT true?

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AIf S_2 is True, then S_1 is TrueBIf S_2 is False, then S_1 is FalseCIf S_1 is False, then S_2 is TrueDIf S_1 is False, then S_2 is False

Step-by-step solutionView

Correct answer

Since $S_{1}$ and $S_{2}$ are logically equivalent, if $S_{1}$ is False, then $S_{2}$ must also be False, making the claim "If $S_{1}$ is False, then $S_{2}$ is True" NOT true.

Option analysis

Why each option works or fails

A · If S_2 is True, then S_1 is True

The student identifies that $S_{1} \equiv S_{2}$ , meaning if $S_{2}$ is True, $S_{1}$ is True, and incorrectly selects this true statement because they missed the word 'NOT' in the question stem. Always re-read the question stem to verify whether you need to select the true statement or the statement that is NOT true.

B · If S_2 is False, then S_1 is False

The student recognizes that $S_{1}$ and $S_{2}$ always share identical truth values, sees that $S_{2} = False ⟹ S_{1} = False$ is valid, but forgets the prompt seeks the false statement. Ensure you check the polarity of the question; finding a valid deduction means it is NOT the correct answer for a 'NOT true' question.

C · If S_1 is False, then S_2 is True

This statement asserts that $S_{1}$ and $S_{2}$ take opposite truth values when $S_{1}$ is False, which directly contradicts their logical equivalence $S_{1} \equiv S_{2}$ . Because $S_{1} \equiv S_{2}$ , whenever $S_{1}$ is False, $S_{2}$ must also be False, which makes this implication false and therefore the correct answer to the question.

D · If S_1 is False, then S_2 is False

The student correctly realizes that if $S_{1}$ is False, then $S_{2}$ must be False, but selects this option believing the goal is to find a true deduction. Watch for the negation in the question stem: an option that is mathematically true is incorrect when asked for the statement that is NOT true.

Reviewed route

Solution

StepWorking

01Concept

Use associative and idempotent laws of disjunction: $(A \lor B) \lor (A \lor C) \equiv A \lor B \lor C$ . Also use the conditional law: $p i g h t a r r o w X \equiv \sim p \lor X$ .

02Option verdict

Since $S_{1} \equiv S_{2}$ , if $S_{2}$ is True, then $S_{1}$ is True. This statement is mathematically true, so it is NOT the correct choice for a question asking for what is NOT true.

03Option verdict

Since $S_{1} \equiv S_{2}$ , if $S_{2}$ is False, then $S_{1}$ is False. This statement is mathematically true, hence not the answer.

04Option verdict

Since $S_{1} \equiv S_{2}$ , whenever $S_{1}$ is False, $S_{2}$ MUST also be False. The assertion 'If $S_{1}$ is False, then $S_{2}$ is True' directly contradicts logical equivalence. Thus, this is NOT true.

05Option verdict

Since $S_{1} \equiv S_{2}$ , if $S_{1}$ is False, then $S_{2}$ is False. This is a true statement, so it is not the answer.

✓Discriminator

Since $S_{1} \equiv S_{2}$ , their truth values must always match in every interpretation. Any option claiming $S_{1}$ and $S_{2}$ have opposite truth values is false.

Hints that build this answer step by step

Using the associative and idempotent laws of disjunction, how can $S_{1} : ((\sim p) \lor q) \lor ((\sim p) \lor r)$ be simplified?
$(\sim p) \lor (q \lor r)$
How can the conditional statement $S_{2} : p \to (q \lor r)$ be rewritten using disjunction and negation?
$(\sim p) \lor (q \lor r)$
Since $S_{1} \equiv S_{2}$ , what relationship holds between their truth values?
$S_{1}$ and $S_{2}$ always share the same truth value in every case.
Based on $S_{1} \equiv S_{2}$ , which of the given options is NOT true?
If $S_{1}$ is False, then $S_{2}$ is True

✓ Source and academic review

Question type: Single correct
Exam relevance: JEE Main · Mathematics
Academic status: Answer verified against the official NTA key
Source: Previous-year question
Editorial review: 22 September 2026

Quick checks

Students also ask

Why is

(\sim p \lor q) \lor (\sim p \lor r)

the same as

\sim p \lor (q \lor r)

Disjunction $\lor$ is associative and commutative, so the expression is simply $\sim p \lor \sim p \lor q \lor r$ . By idempotence, $\sim p \lor \sim p \equiv \sim p$ , yielding $\sim p \lor (q \lor r)$ .

Answer

Since $S_{1}$ and $S_{2}$ are logically equivalent, if $S_{1}$ is False, then $S_{2}$ must also be False, making the claim "If $S_{1}$ is False, then $S_{2}$ is True" NOT true.

Why each option works or fails

A: If S_2 is True, then S_1 is True - The student identifies that $S_{1} \equiv S_{2}$ , meaning if $S_{2}$ is True, $S_{1}$ is True, and incorrectly selects this true statement because they missed the word 'NOT' in the question stem. Always re-read the question stem to verify whether you need to select the true statement or the statement that is NOT true.
B: If S_2 is False, then S_1 is False - The student recognizes that $S_{1}$ and $S_{2}$ always share identical truth values, sees that $S_{2} = False ⟹ S_{1} = False$ is valid, but forgets the prompt seeks the false statement. Ensure you check the polarity of the question; finding a valid deduction means it is NOT the correct answer for a 'NOT true' question.
C · correct: If S_1 is False, then S_2 is True - This statement asserts that $S_{1}$ and $S_{2}$ take opposite truth values when $S_{1}$ is False, which directly contradicts their logical equivalence $S_{1} \equiv S_{2}$ . Because $S_{1} \equiv S_{2}$ , whenever $S_{1} \equiv S_{2}$ is False, $S_{2}$ must also be False, which makes this implication false and therefore the correct answer to the question.
D: If S_1 is False, then S_2 is False - The student correctly realizes that if $S_{1}$ is False, then $S_{2}$ must be False, but selects this option believing the goal is to find a true deduction. Watch for the negation in the question stem: an option that is mathematically true is incorrect when asked for the statement that is NOT true.

Step-by-step solution

Concept: Use associative and idempotent laws of disjunction: $(A \lor B) \lor (A \lor C) \equiv A \lor B \lor C$ . Also use the conditional law: $p i g h t a r r o w X \equiv \sim p \lor X$ .
Option verdict: Since $S_{1} \equiv S_{2}$ , if $S_{2}$ is True, then $S_{1}$ is True. This statement is mathematically true, so it is NOT the correct choice for a question asking for what is NOT true.
Option verdict: Since $S_{1} \equiv S_{2}$ , if $S_{2}$ is False, then $S_{1}$ is False. This statement is mathematically true, hence not the answer.
Option verdict: Since $S_{1} \equiv S_{2}$ , whenever $S_{1}$ is False, $S_{2}$ MUST also be False. The assertion 'If $S_{1}$ is False, then $S_{2}$ is True' directly contradicts logical equivalence. Thus, this is NOT true.
Option verdict: Since $S_{1} \equiv S_{2}$ , if $S_{1}$ is False, then $S_{2}$ is False. This is a true statement, so it is not the answer.
Discriminator: Since $S_{1} \equiv S_{2}$ , their truth values must always match in every interpretation. Any option claiming $S_{1}$ and $S_{2}$ have opposite truth values is false.

Logical Equivalence of Disjunctive and Conditional Compound Statements

What feels right?

Since $S_{1}$ and $S_{2}$ are logically equivalent, if $S_{1}$ is False, then $S_{2}$ must also be False, making the claim "If $S_{1}$ is False, then $S_{2}$ is True" NOT true.

Why each option works or fails

Solution

01Concept

02Option verdict

03Option verdict

04Option verdict

05Option verdict

✓Discriminator

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Students also ask

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01Concept

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The Boolean expression $(\sim (p \land q)) \lor q$ is equivalent to…

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