Negation of the statement `p implies (~q ^^r)` is

Options

A. `p ^^ (q vv ~ r)`
B. `~ p (^^ (q vv ~r )`
C. `p vv (q vv ~ r)`
D. None of these

Answer

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

### Step 1: Rewrite the Implication
The implication \( p \implies (\neg q \land r) \) can be rewritten using the equivalence \( p \implies q \equiv \neg p \lor q \). Thus, we have:
\[ p \implies (\neg q \land r) \equiv \neg p \lor (\neg q \land r) \]

### Step 2: Apply De Morgan's Law
Next, we need to find the negation of the entire expression:
\[ \neg(p \implies (\neg q \land r)) \equiv \neg(\neg p \lor (\neg q \land r)) \]
Using De Morgan's Law, we can distribute the negation:
\[ \neg(\neg p \lor (\neg q \land r)) \equiv \neg(\neg p) \land \neg(\neg q \land r) \]
This simplifies to:
\[ p \land \neg(\neg q \land r) \]

### Step 3: Apply De Morgan's Law Again
Now, we apply De Morgan's Law to the second part:
\[ \neg(\neg q \land r) \equiv \neg(\neg q) \lor \neg r \equiv q \lor \neg r \]
Thus, we have:
\[ p \land (q \lor \neg r) \]

### Step 4: Final Expression
The final expression for the negation of the original statement is:
\[ p \land (q \lor \neg r) \]

### Summary
The negation of the statement \( p \implies (\neg q \land r) \) is:
\[ p \land (q \lor \neg r) \]

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