Negation of the statement `pto(q^^r)` is

Options

A. `pto~(q^^r)`
B. `~pvv(q^^r)`
C. `p^^(~p^^~r)`
D. `(q^^r)top`

Explanation

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

### Step 1: Understand the implication
The implication \( p \to (q \land r) \) can be rewritten using logical equivalence. The implication \( p \to A \) is equivalent to \( \neg p \lor A \). Therefore, we can rewrite the statement as:
\[ p \to (q \land r) \equiv \neg p \lor (q \land r) \]

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

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

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

### Final Answer
Putting it all together, the negation of the statement \( p \to (q \land r) \) is:
\[ p \land (\neg q \lor \neg r) \]