Find the negation of `qvv ~(p ^^q) to r.`

Answer

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

### Step 1: Rewrite the Implication
The implication \( A \to B \) can be rewritten using logical equivalence as \( \neg A \lor B \). Thus, we can rewrite our statement as:
\[ \neg(q \lor \neg(p \land q)) \lor r \]

### Step 2: Apply De Morgan's Law
Next, we will apply De Morgan's Law to the negation of the disjunction:
\[ \neg(q \lor \neg(p \land q)) = \neg q \land \neg(\neg(p \land q)) \]
This simplifies to:
\[ \neg q \land (p \land q) \]

### Step 3: Combine the Expressions
Now we substitute this back into our expression:
\[ (\neg q \land (p \land q)) \lor r \]

### Step 4: Negate the Entire Expression
Now we need to negate the entire expression:
\[ \neg((\neg q \land (p \land q)) \lor r) \]
Using De Morgan's Law again, we get:
\[ \neg(\neg q \land (p \land q)) \land \neg r \]

### Step 5: Apply De Morgan's Law Again
Now we will apply De Morgan's Law to the first part:
\[ \neg(\neg q \land (p \land q)) = \neg(\neg q) \lor \neg(p \land q) = q \lor (\neg p \lor \neg q) \]
This simplifies to:
\[ q \lor \neg p \lor \neg q \]

### Step 6: Combine the Results
Now we combine this with the negation of \( r \):
\[ (q \lor \neg p \lor \neg q) \land \neg r \]

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

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