The negation of the expression `q vv((~q)^^p)` is equivalent to

Options

A. `p^^(~q)`
B. `(~p)^^(~q)`
C. `(~p)vv(~q)`
D. `(~p)vv q`

Answer

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

### Step 1: Write the negation of the expression
We start with the expression:
\[ \neg (q \lor (\neg q \land p)) \]

### Step 2: Apply De Morgan's Law
According to De Morgan's Law, the negation of a disjunction is the conjunction of the negations. Thus, we can rewrite the expression as:
\[ \neg q \land \neg (\neg q \land p) \]

### Step 3: Apply De Morgan's Law again
Now we need to simplify \(\neg (\neg q \land p)\). Again, applying De Morgan's Law:
\[ \neg (\neg q \land p) = \neg (\neg q) \lor \neg p = q \lor \neg p \]
So now we have:
\[ \neg q \land (q \lor \neg p) \]

### Step 4: Distribute \(\neg q\)
Next, we can distribute \(\neg q\) across the expression:
\[ (\neg q \land q) \lor (\neg q \land \neg p) \]

### Step 5: Simplify the expression
Notice that \(\neg q \land q\) is always false (a contradiction), so we can simplify:
\[ \text{False} \lor (\neg q \land \neg p) = \neg q \land \neg p \]

### Conclusion
Thus, the negation of the expression \( q \lor (\neg q \land p) \) is equivalent to:
\[ \neg q \land \neg p \]

### Final Answer
The negation of the expression \( q \lor (\neg q \land p) \) is equivalent to \( \neg p \land \neg q \).

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