Q61 / 150 · MHT CET 2026 (20 April Shift 1) · Mathematics
The negation of the inverse of the statement $\sim p \vee \sim q$ is...
- A$p \wedge q$
- B$\sim p \wedge q$
- Correct answerC$\sim p \wedge \sim q$
- D$p \vee q$
Solution
The given statement is $\sim p \vee \sim q$.
We know that a disjunction $A \vee B$ can be expressed as a conditional statement $\sim A \rightarrow B$.
Thus, the statement $\sim p \vee \sim q$ can be written as $p \rightarrow \sim q$.
The inverse of a conditional statement $A \rightarrow B$ is $\sim A \rightarrow \sim B$.
Therefore, the inverse of $p \rightarrow \sim q$ is $\sim p \rightarrow \sim(\sim q)$, which simplifies to $\sim p \rightarrow q$.
Converting this inverse back to a logical disjunction, we get $\sim(\sim p) \vee q \equiv p \vee q$.
The negation of this inverse is $\sim(p \vee q)$.
By De Morgan's Law, $\sim(p \vee q) \equiv \sim p \wedge \sim q$.
Answer: $\sim p \wedge \sim q$
We know that a disjunction $A \vee B$ can be expressed as a conditional statement $\sim A \rightarrow B$.
Thus, the statement $\sim p \vee \sim q$ can be written as $p \rightarrow \sim q$.
The inverse of a conditional statement $A \rightarrow B$ is $\sim A \rightarrow \sim B$.
Therefore, the inverse of $p \rightarrow \sim q$ is $\sim p \rightarrow \sim(\sim q)$, which simplifies to $\sim p \rightarrow q$.
Converting this inverse back to a logical disjunction, we get $\sim(\sim p) \vee q \equiv p \vee q$.
The negation of this inverse is $\sim(p \vee q)$.
By De Morgan's Law, $\sim(p \vee q) \equiv \sim p \wedge \sim q$.
Answer: $\sim p \wedge \sim q$
Hence, the correct answer is option: 'C'.