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

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the Proposition**: The given proposition is \( q \lor \neg (p \land r) \).

2. **Apply Negation**: We need to find the negation of the entire proposition, which is \( \neg (q \lor \neg (p \land r)) \).

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

4. **Simplify the Negation**: The negation of a negation cancels out, so we have:
\[ \neg q \land (p \land r) \]

5. **Rearranging the Expression**: We can rearrange the expression using the associative property of conjunction:
\[ \neg q \land p \land r \]

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