Negation of the statement ` ~ p to (q vv r)` is

Options

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

Explanation

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

### Step 1: Understand the implication
The statement \( \neg p \to (q \lor r) \) can be rewritten using the implication equivalence:
\[ A \to B \equiv \neg A \lor B \]
Here, \( A \) is \( \neg p \) and \( B \) is \( q \lor r \). Thus, we can rewrite the statement as:
\[ \neg(\neg p) \lor (q \lor r) \]
This simplifies to:
\[ p \lor (q \lor r) \]

### Step 2: Negate the entire statement
Now we need to find the negation of the expression \( p \lor (q \lor r) \):
\[ \neg(p \lor (q \lor r)) \]

### Step 3: Apply De Morgan's Laws
Using De Morgan's laws, which state that:
\[ \neg(A \lor B) \equiv \neg A \land \neg B \]
we can apply this to our expression:
\[ \neg(p \lor (q \lor r)) \equiv \neg p \land \neg(q \lor r) \]

### Step 4: Further simplify using De Morgan's Laws
Next, we need to simplify \( \neg(q \lor r) \) using De Morgan's laws again:
\[ \neg(q \lor r) \equiv \neg q \land \neg r \]
Thus, we can substitute this back into our expression:
\[ \neg p \land (\neg q \land \neg r) \]

### Step 5: Combine the results
This can be rewritten as:
\[ \neg p \land \neg q \land \neg r \]

### Final Result
So, the negation of the statement \( \neg p \to (q \lor r) \) is:
\[ \neg p \land \neg q \land \neg r \]