If the inverse of implication ` p to q ` is defined as ` ~ p to ~q` , then the inverse of the proposition ` ( p ^^ ~ q) to r ` is

Options

A. `~r to (~pvvq)`
B. `r to (p^^~q)`
C. `~qvv(p^^r)`
D. none of these

Explanation

To find the inverse of the proposition \( (p \land \neg q) \to r \), we will follow the steps outlined below:

### Step 1: Identify the given proposition
The given proposition is \( (p \land \neg q) \to r \).

### Step 2: Apply the definition of the inverse
According to the definition provided, the inverse of an implication \( p \to q \) is \( \neg p \to \neg q \). Therefore, we need to negate both the antecedent and the consequent of our given proposition.

### Step 3: Negate the antecedent and the consequent
The antecedent of our implication is \( (p \land \neg q) \) and the consequent is \( r \). Thus, we need to negate both:
- Negation of the antecedent: \( \neg(p \land \neg q) \)
- Negation of the consequent: \( \neg r \)

### Step 4: Write the inverse proposition
Now we can write the inverse of the proposition:
\[ \neg(p \land \neg q) \to \neg r \]

### Step 5: Apply De Morgan's Law
Using De Morgan's Law, we can simplify \( \neg(p \land \neg q) \):
\[ \neg(p \land \neg q) = \neg p \lor \neg(\neg q) = \neg p \lor q \]
Thus, we can rewrite our inverse proposition as:
\[ (\neg p \lor q) \to \neg r \]

### Final Answer
The inverse of the proposition \( (p \land \neg q) \to r \) is:
\[ (\neg p \lor q) \to \neg r \]