The negation of the statement `(p vv q) implies (q vvr)`

Options

A. `p wedge q wedge r`
B. `p wedge q wedge ~r`
C. `p wedge ~q wedge ~r`
D. `~p wedge q wedge r`

Answer

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

### Step 1: Understand the implication
The statement \((p \lor q) \implies (q \lor r)\) can be rewritten using the definition of implication. An implication \(A \implies B\) is logically equivalent to \(\neg A \lor B\). Therefore, we can rewrite the statement as:
\[ \neg(p \lor q) \lor (q \lor r) \]

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

### Step 3: Apply De Morgan's Law
Using De Morgan's Law, we know that \(\neg(A \lor B) = \neg A \land \neg B\). Thus, we can apply this to our expression:
\[ \neg(\neg(p \lor q)) \land \neg(q \lor r) \]

### Step 4: Simplify the negation
The negation of \(\neg(p \lor q)\) simplifies to \(p \lor q\). The negation of \((q \lor r)\) can also be simplified using De Morgan's Law:
\[ \neg(q \lor r) = \neg q \land \neg r \]
So our expression now looks like:
\[ (p \lor q) \land (\neg q \land \neg r) \]

### Step 5: Combine the expressions
We can combine the expressions:
\[ (p \lor q) \land \neg q \land \neg r \]

### Step 6: Distribute the conjunction
Distributing the conjunction gives us:
\[ (p \land \neg q \land \neg r) \lor (q \land \neg q \land \neg r) \]
However, \(q \land \neg q\) is always false, so we can ignore that part:
\[ p \land \neg q \land \neg r \]

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