The contrapositive of `(~p ^^ q) rarr (q ^^ ~r)` is

Options

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

Answer

To find the contrapositive of the statement \((\neg p \land q) \implies (q \land \neg r)\), we will follow these steps:

### Step 1: Identify the original statement
The original statement is:
\[ (\neg p \land q) \implies (q \land \neg r) \]

### Step 2: Recall the definition of contrapositive
The contrapositive of a statement of the form \(A \implies B\) is \(\neg B \implies \neg A\).

### Step 3: Identify \(A\) and \(B\)
In our case:
- \(A\) is \((\neg p \land q)\)
- \(B\) is \((q \land \neg r)\)

### Step 4: Negate \(B\)
To find \(\neg B\):
\[ B = (q \land \neg r) \]
Using De Morgan's laws, we have:
\[ \neg B = \neg(q \land \neg r) = \neg q \lor r \]

### Step 5: Negate \(A\)
Now, we find \(\neg A\):
\[ A = (\neg p \land q) \]
Again, using De Morgan's laws:
\[ \neg A = \neg(\neg p \land q) = p \lor \neg q \]

### Step 6: Form the contrapositive
Now we can write the contrapositive:
\[ \neg B \implies \neg A \]
Substituting \(\neg B\) and \(\neg A\):
\[ (\neg q \lor r) \implies (p \lor \neg q) \]

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