The negative of ` p ^^ ~ ( ~ q ^^ r)` is

Options

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

Answer

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

### Step 1: Write the expression
The original expression is:
\[ p \land \neg(\neg q \land r) \]

### Step 2: Apply De Morgan's Law
To find the negation, we apply De Morgan's Law, which states that the negation of a conjunction is the disjunction of the negations. Thus, we have:
\[ \neg(p \land \neg(\neg q \land r)) = \neg p \lor \neg(\neg(\neg q \land r)) \]

### Step 3: Simplify the inner negation
Now we need to simplify the inner negation:
\[ \neg(\neg q \land r) \]
Applying De Morgan's Law again, we get:
\[ \neg(\neg q \land r) = \neg(\neg q) \lor \neg r = q \lor \neg r \]

### Step 4: Substitute back into the expression
Now substitute this back into our expression:
\[ \neg p \lor (q \lor \neg r) \]

### Step 5: Rearrange the expression
We can rearrange this expression for clarity:
\[ \neg p \lor q \lor \neg r \]

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