`"The negation of the statement =" [(~p^^q)vv(p^^~q)]` `" is "`

Options

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

Answer

To find the negation of the statement \[(\neg p \land q) \lor (p \land \neg q)\], we will follow the steps of logical negation and apply De Morgan's laws.

### Step-by-Step Solution:

1. **Write down the original statement**:
\[ S = (\neg p \land q) \lor (p \land \neg q) \]

2. **Apply negation to the entire statement**:
\[ \neg S = \neg[(\neg p \land q) \lor (p \land \neg q)] \]

3. **Use De Morgan's Law**: According to De Morgan's laws, the negation of a disjunction is the conjunction of the negations:
\[ \neg S = \neg(\neg p \land q) \land \neg(p \land \neg q) \]

4. **Apply De Morgan's Law to each part**:
- For the first part \(\neg(\neg p \land q)\):
\[ \neg(\neg p \land q) = \neg(\neg p) \lor \neg(q) = p \lor \neg q \]
- For the second part \(\neg(p \land \neg q)\):
\[ \neg(p \land \neg q) = \neg(p) \lor \neg(\neg q) = \neg p \lor q \]

5. **Combine the results**:
\[ \neg S = (p \lor \neg q) \land (\neg p \lor q) \]

6. **Final expression**:
The negation of the original statement is:
\[ \neg S = (p \lor \neg q) \land (\neg p \lor q) \]