The nagation of statement `(p ^^q) vv ( q vv ~r)`

Options

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

Answer

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

### Step 1: Identify the original statement
The original statement is:
\[ (p \land q) \lor (q \lor \neg r) \]

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

### 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 have:
\[ \neg((p \land q) \lor (q \lor \neg r)) = \neg(p \land q) \land \neg(q \lor \neg r) \]

### Step 4: Negate each part
Now we need to negate each part:
1. For \(\neg(p \land q)\), we apply De Morgan's Laws again:
\[ \neg(p \land q) = \neg p \lor \neg q \]
2. For \(\neg(q \lor \neg r)\), we apply De Morgan's Laws:
\[ \neg(q \lor \neg r) = \neg q \land r \]

### Step 5: Combine the results
Now we combine the results from the negations:
\[ \neg((p \land q) \lor (q \lor \neg r)) = (\neg p \lor \neg q) \land (\neg q \land r) \]

### Step 6: Simplify the expression
We can simplify the expression further:
\[ (\neg p \lor \neg q) \land \neg q \land r \]
This can be rearranged as:
\[ \neg p \lor (\neg q \land r) \]

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

---