The negation of `p ^^ (q to ~ r)` is

Options

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

Explanation

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

### Step 1: Write down the original expression
The original expression we need to negate is:
\[ p \land (q \to \neg r) \]

### Step 2: Apply the negation
To negate the expression, we apply the negation operator:
\[ \neg (p \land (q \to \neg r)) \]

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

### Step 4: Rewrite the implication
The implication \( q \to \neg r \) can be rewritten using its logical equivalence:
\[ q \to \neg r \equiv \neg q \lor \neg r \]
Thus, we can substitute this into our expression:
\[ \neg (q \to \neg r) = \neg (\neg q \lor \neg r) \]

### Step 5: Apply De Morgan's Law again
Now, we apply De Morgan's Law to the negation of the disjunction:
\[ \neg (\neg q \lor \neg r) = q \land r \]

### Step 6: Combine the results
Now we can substitute this back into our expression from Step 3:
\[ \neg p \lor (q \land r) \]

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