`PDeltaQ=P+Q, if PQ lt0`
`PDeltaQ=P-Q, if PQ gt0`
`[(2Delta3)Delta(-2Delta3)]` is equal to :

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the question step by step, we will evaluate the expression \( (2 \Delta 3) \Delta (-2 \Delta 3) \) using the defined operations for \( \Delta \).

### Step 1: Calculate \( 2 \Delta 3 \)
According to the definition:
- If \( P \cdot Q > 0 \), then \( P \Delta Q = P - Q \).
- If \( P \cdot Q < 0 \), then \( P \Delta Q = P + Q \).

Here, \( P = 2 \) and \( Q = 3 \).
- Calculate \( P \cdot Q = 2 \cdot 3 = 6 \), which is greater than 0.
- Therefore, we use the operation \( P - Q \):
\[ 2 \Delta 3 = 2 - 3 = -1 \]

### Step 2: Calculate \( -2 \Delta 3 \)
Now we need to find \( -2 \Delta 3 \).
Here, \( P = -2 \) and \( Q = 3 \).
- Calculate \( P \cdot Q = -2 \cdot 3 = -6 \), which is less than 0.
- Therefore, we use the operation \( P + Q \):
\[ -2 \Delta 3 = -2 + 3 = 1 \]

### Step 3: Calculate \( (2 \Delta 3) \Delta (-2 \Delta 3) \)
Now we substitute the values we found into the expression:
\[ (2 \Delta 3) \Delta (-2 \Delta 3) = (-1) \Delta 1 \]
Here, \( P = -1 \) and \( Q = 1 \).
- Calculate \( P \cdot Q = -1 \cdot 1 = -1 \), which is less than 0.
- Therefore, we use the operation \( P + Q \):
\[ -1 \Delta 1 = -1 + 1 = 0 \]

### Final Answer
The final answer to the expression \( (2 \Delta 3) \Delta (-2 \Delta 3) \) is:
\[ \boxed{0} \]