The inverse of the proposition `(p ^^ ~q) rarr r` is

Options

A. `~r rarr ~p vv q`
B. `~p vv q rarr ~r`
C. `r rarr p ^^ ~q`
D. `~q vv r rarr p`

Answer

To find the inverse of the proposition \( (p \land \neg q) \implies r \), we will follow the definition of the inverse of a proposition. The inverse of a proposition \( A \implies B \) is given by \( \neg A \implies \neg B \).

### Step-by-Step Solution:

1. **Identify the Proposition**:
We have the proposition \( (p \land \neg q) \implies r \).

2. **Identify A and B**:
Here, let \( A = (p \land \neg q) \) and \( B = r \).

3. **Apply the Inverse Definition**:
The inverse of \( A \implies B \) is \( \neg A \implies \neg B \).

4. **Negate A**:
We need to find \( \neg A \):
\[ \neg A = \neg (p \land \neg q) \]
Using De Morgan's laws, this can be rewritten as:
\[ \neg A = \neg p \lor \neg (\neg q) = \neg p \lor q \]

5. **Negate B**:
Now we find \( \neg B \):
\[ \neg B = \neg r \]

6. **Combine Negations**:
Now we can write the inverse:
\[ \neg A \implies \neg B \implies (\neg p \lor q) \implies \neg r \]

Thus, the inverse of the proposition \( (p \land \neg q) \implies r \) is:
\[ (\neg p \lor q) \implies \neg r \]

### Final Answer:
The inverse of the proposition \( (p \land \neg q) \implies r \) is \( (\neg p \lor q) \implies \neg r \).

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