`(~(pvvq))vv(~p^^q)` is logically equivalent to

Options

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

Explanation

To determine whether the expression \((\neg (p \lor q)) \lor (\neg p \land q)\) is logically equivalent to another expression, we can simplify it step by step.

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ (\neg (p \lor q)) \lor (\neg p \land q) \]

2. **Apply De Morgan's Law**:
According to De Morgan's laws, the negation of a disjunction is the conjunction of the negations. Thus:
\[ \neg (p \lor q) = \neg p \land \neg q \]
Substituting this into our expression gives:
\[ (\neg p \land \neg q) \lor (\neg p \land q) \]

3. **Factor Out Common Terms**:
Notice that \(\neg p\) is common in both terms:
\[ \neg p \land (\neg q \lor q) \]

4. **Simplify Using the Law of Excluded Middle**:
The expression \((\neg q \lor q)\) is always true (T) because either \(q\) is true or it is false. Therefore:
\[ \neg p \land T = \neg p \]

5. **Final Result**:
Thus, the original expression simplifies to:
\[ \neg p \]

### Conclusion:
The expression \((\neg (p \lor q)) \lor (\neg p \land q)\) is logically equivalent to \(\neg p\).