The Naegation of the statement `(p ^^q) to r` is-

Options

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

Answer

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

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

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

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

### Step 4: Simplify the negation
The negation of \( \neg (p \land q) \) is simply \( p \land q \). Therefore, we can simplify our expression:
\[ (p \land q) \land \neg r \]

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