The contrapositive of `p implies (~p ^^ q)` is-

Options

A. `~p implies (p vv ~q)`
B. `(p ^^ ~q) implies ~p`
C. `(p vv~q) impliesp`
D. `(p vv ~q) implies ~p`

Answer

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

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

### Step 2: Understand the contrapositive
The contrapositive of a statement \( p \implies q \) is \( \neg q \implies \neg p \). This means we need to find \( \neg (\neg p \land q) \) and \( \neg p \).

### Step 3: Negate the conclusion
We need to negate the conclusion \( \neg p \land q \). Using De Morgan's laws, we have:
\[ \neg (\neg p \land q) = \neg (\neg p) \lor \neg q = p \lor \neg q \]

### Step 4: Write the contrapositive
Now, we can write the contrapositive using the negation we found:
\[ \neg (\neg p \land q) \implies \neg p \]
This translates to:
\[ (p \lor \neg q) \implies \neg p \]

### Step 5: Finalize the contrapositive
Thus, the contrapositive of the original statement \( p \implies (\neg p \land q) \) is:
\[ p \lor \neg q \implies \neg p \]

### Conclusion
The correct answer is:
\[ p \lor \neg q \implies \neg p \]