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. `~p rarr (q ^^ r)`

Answer

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

### Step 1: Understand the Inverse
The inverse of a proposition of the form \(P \implies Q\) is given by \(\neg P \implies \neg Q\).

### Step 2: Identify \(P\) and \(Q\)
In our case, we have:
- \(P\) as \((p \land \neg q)\)
- \(Q\) as \(r\)

### Step 3: Apply the Inverse
Using the definition of the inverse, we can write:
\[ \text{Inverse} = \neg (p \land \neg q) \implies \neg r \]

### Step 4: Simplify \(\neg (p \land \neg q)\)
Using De Morgan's laws, we can simplify \(\neg (p \land \neg q)\):
\[ \neg (p \land \neg q) = \neg p \lor \neg (\neg q) = \neg p \lor q \]

### Step 5: Substitute Back into the Inverse
Now substituting back into our inverse expression, we have:
\[ \neg p \lor q \implies \neg r \]

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

### Conclusion
The final answer is \((\neg p \lor q) \implies \neg r\).

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