The negation of `(~p ^^ q) vv (p ^^ ~ q)` is

Options

A. `(pvv~q)^^(~pvvq)`
B. `(p^^~q)^^(~pvvq)`
C. `(p^^~q)^^(pvv~q)`
D. both `SO_(4)^(2-)` and `NO_(3)^(-)`

Answer

To find the negation of the expression \((\neg p \land q) \lor (p \land \neg q)\), we will follow the logical negation rules step by step.

### Step-by-Step Solution:

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

2. **Apply Negation**:
We want to find the negation of the entire expression:
\[ \neg((\neg p \land q) \lor (p \land \neg q)) \]

3. **Use De Morgan's Laws**:
According to De Morgan's Laws, the negation of an OR statement is the AND of the negations:
\[ \neg(A \lor B) = \neg A \land \neg B \]
Here, let \(A = \neg p \land q\) and \(B = p \land \neg q\). Thus, we have:
\[ \neg((\neg p \land q) \lor (p \land \neg q)) = \neg(\neg p \land q) \land \neg(p \land \neg q) \]

4. **Negate Each Component**:
Now we need to negate each of the components:
- For \(\neg(\neg p \land q)\), we again use De Morgan's Laws:
\[ \neg(\neg p \land q) = \neg(\neg p) \lor \neg q = p \lor \neg q \]
- For \(\neg(p \land \neg q)\), we apply De Morgan's Laws:
\[ \neg(p \land \neg q) = \neg p \lor \neg(\neg q) = \neg p \lor q \]

5. **Combine the Results**:
Now we combine the results from the negations:
\[ (p \lor \neg q) \land (\neg p \lor q) \]

6. **Final Expression**:
The final expression for the negation of \((\neg p \land q) \lor (p \land \neg q)\) is:
\[ (p \lor \neg q) \land (\neg p \lor q) \]

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