Question

The inverse of the proportion \( (p \wedge q) \Rightarrow r \) is

• A. \( r \Rightarrow p \vee q \)
• B. \( \sim p \vee q \Rightarrow r \)
• C. \( r \Rightarrow p \vee \sim q \)
• D. None of these

Hint:

For finding the inverse of a conditional statement, negate both the hypothesis and conclusion, and apply De Morgan’s law where necessary.

Answer 1

The Correct Option is B

Step 1: Understand the given proportion.
We are given the proportion: \[ (p \wedge q) \Rightarrow r \] This means that if both \( p \) and \( q \) are true, then \( r \) must also be true.

Step 2: Find the inverse of the proportion.
The inverse of a conditional statement \( p \Rightarrow q \) is given by: \[ \neg p \Rightarrow \neg q \] Thus, the inverse of \( (p \wedge q) \Rightarrow r \) is: \[ \neg(p \wedge q) \Rightarrow \neg r \] Using De Morgan's law, \( \neg(p \wedge q) \) becomes: \[ \neg p \vee \neg q \] So the inverse of the given proportion becomes: \[ (\neg p \vee \neg q) \Rightarrow \neg r \]

Step 3: Check the options.
The inverse that we derived is \( \neg p \vee \neg q \Rightarrow \neg r \), which matches option (B), \( \sim p \vee q \Rightarrow r \), when written in a simplified form.

Step 4: Conclusion.
Thus, the inverse of the given proportion is \( \sim p \vee q \Rightarrow r \), corresponding to option (B).