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

Options

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

Answer

To solve the problem of finding the negative of \( p \cap \neg (p \cap r) \), we will use De Morgan's Laws and properties of negation in set theory. Here’s a step-by-step solution:

### Step 1: Write the expression
We start with the expression we need to negate:
\[ \neg (p \cap \neg (p \cap r)) \]

### Step 2: Apply De Morgan's Law
According to De Morgan's Laws, the negation of an intersection can be expressed as the union of the negations. Therefore, we can rewrite the expression as:
\[ \neg (p \cap \neg (p \cap r)) = \neg p \cup \neg (\neg (p \cap r)) \]

### Step 3: Simplify the double negation
The double negation \( \neg (\neg (p \cap r)) \) simplifies to \( p \cap r \). Thus, we can rewrite our expression:
\[ \neg p \cup (p \cap r) \]

### Step 4: Final expression
The final expression for the negation of \( p \cap \neg (p \cap r) \) is:
\[ \neg p \cup (p \cap r) \]

### Conclusion
The negative of \( p \cap \neg (p \cap r) \) is:
\[ \neg p \cup (p \cap r) \]