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

Options

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

Answer

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

### Step 1: Understand the Given Statement
The statement we need to negate is:
\[ S = p \land (q \rightarrow \neg r) \]

### Step 2: Rewrite the Implication
Recall that an implication \( q \rightarrow \neg r \) can be rewritten using logical equivalence:
\[ q \rightarrow \neg r \equiv \neg q \lor \neg r \]
Thus, we can rewrite \( S \) as:
\[ S = p \land (\neg q \lor \neg r) \]

### Step 3: Apply De Morgan's Law
To find the negation of \( S \), we apply De Morgan's Law:
\[ \neg S = \neg (p \land (\neg q \lor \neg r)) \]
According to De Morgan's Law, this can be expressed as:
\[ \neg S = \neg p \lor \neg (\neg q \lor \neg r) \]

### Step 4: Simplify the Negation
Next, we simplify \( \neg (\neg q \lor \neg r) \) using De Morgan's Law again:
\[ \neg (\neg q \lor \neg r) = q \land r \]
Thus, we can rewrite \( \neg S \) as:
\[ \neg S = \neg p \lor (q \land r) \]

### Final Result
The negation of the statement \( p \land (q \rightarrow \neg r) \) is:
\[ \neg S = \neg p \lor (q \land r) \]