For any two statements p and q, the negation of the expression `p vv(~p ^^ q)` is :

Options

A. `~p^^ ~q`
B. `P^^q`
C. `pharrq`
D. `~pvv~q`

Answer

To find the negation of the expression \( p \lor (\neg p \land q) \), we will follow these steps:

### Step 1: Identify the expression
The expression we need to negate is:
\[ p \lor (\neg p \land q) \]

### Step 2: Apply De Morgan's Law
To negate the expression, we apply De Morgan's Law, which states that the negation of a disjunction is the conjunction of the negations:
\[ \neg (A \lor B) = \neg A \land \neg B \]
In our case, let \( A = p \) and \( B = \neg p \land q \). Thus, we have:
\[ \neg (p \lor (\neg p \land q)) = \neg p \land \neg (\neg p \land q) \]

### Step 3: Negate the second part
Now we need to negate the second part \( \neg p \land q \). Again, applying De Morgan's Law:
\[ \neg (C \land D) = \neg C \lor \neg D \]
where \( C = \neg p \) and \( D = q \). Therefore:
\[ \neg (\neg p \land q) = \neg (\neg p) \lor \neg q = p \lor \neg q \]

### Step 4: Combine the results
Now we substitute back into our expression:
\[ \neg (p \lor (\neg p \land q)) = \neg p \land (p \lor \neg q) \]

### Step 5: Distribute the terms
Now we can distribute \( \neg p \) across \( (p \lor \neg q) \):
\[ \neg p \land (p \lor \neg q) = (\neg p \land p) \lor (\neg p \land \neg q) \]

### Step 6: Simplify the expression
The term \( \neg p \land p \) is always false (contradiction), so we can simplify:
\[ \neg p \land (p \lor \neg q) = \text{False} \lor (\neg p \land \neg q) = \neg p \land \neg q \]

### Final Result
Thus, the negation of the expression \( p \lor (\neg p \land q) \) is:
\[ \neg p \land \neg q \]