Negation of `pvv(q^^~p)` is

Options

A. `p^^q`
B. `~pvv~q`
C. `~p vvq`
D. `~p ^^ ~q`

Answer

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

### Step 1: Write down the original statement
The original statement is:
\[ p \lor (q \land \neg p) \]

### Step 2: Apply the negation
We need to find the negation of the entire statement:
\[ \neg (p \lor (q \land \neg p)) \]

### Step 3: Use De Morgan's Law
According to De Morgan's Law, the negation of a disjunction is the conjunction of the negations:
\[ \neg (p \lor (q \land \neg p)) = \neg p \land \neg (q \land \neg p) \]

### Step 4: Apply De Morgan's Law again
Now, we need to negate the second part \( \neg (q \land \neg p) \):
\[ \neg (q \land \neg p) = \neg q \lor p \]
Thus, we can rewrite our expression:
\[ \neg p \land (\neg q \lor p) \]

### Step 5: Final expression
Putting it all together, we have:
\[ \neg p \land (\neg q \lor p) \]

This is the negation of the original statement \( p \lor (q \land \neg p) \).

### Final Answer
The negation of \( p \lor (q \land \neg p) \) is:
\[ \neg p \land (\neg q \lor p) \]

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