To solve the problem, we need to analyze the sets defined in the question and perform the required operations on them. Let's break it down step by step.
### Step 1: Define the Sets
1. **Set \( Z_N \)**: This is the set of non-negative integers, which includes \( \{0, 1, 2, 3, \ldots\} \).
2. **Set \( Z_P \)**: This is the set of non-positive integers, which includes \( \{ \ldots, -3, -2, -1, 0\} \).
3. **Set \( Z \)**: This is the set of all integers, which includes both positive and negative integers, as well as zero.
4. **Set \( E \)**: This is the set of even integers, which includes \( \{ \ldots, -4, -2, 0, 2, 4, \ldots\} \).
5. **Set \( P \)**: This is the set of prime numbers, which includes \( \{2, 3, 5, 7, 11, \ldots\} \).
### Step 2: Analyze Each Option
#### Option 1: \( E \cap P \)
- **Intersection of even integers and prime numbers**: The only even prime number is \( 2 \).
- Therefore, \( E \cap P = \{2\} \).
- This is not an empty set, so this option is incorrect.
**Hint**: Check for common elements between the two sets.
#### Option 2: \( Z_N \cap Z_P \)
- **Intersection of non-negative integers and non-positive integers**: The only integer that is both non-negative and non-positive is \( 0 \).
- Therefore, \( Z_N \cap Z_P = \{0\} \).
- This is a singleton set, so this option is incorrect.
**Hint**: Identify elements that belong to both sets.
#### Option 3: \( Z - Z_N \)
- **Subtracting non-negative integers from all integers**: This means we are left with negative integers, which is \( Z_P \).
- Therefore, \( Z - Z_N = Z_P \).
- This option is correct.
**Hint**: Understand what remains after removing a subset from a larger set.
#### Option 4: Symmetric Difference \( Z_N \Delta Z_P \)
- **Symmetric difference**: This is defined as \( (Z_N \cup Z_P) - (Z_N \cap Z_P) \).
- **Union of \( Z_N \) and \( Z_P \)**: This includes all integers (both positive and negative).
- **Intersection of \( Z_N \) and \( Z_P \)**: As established, this is \( \{0\} \).
- Therefore, the symmetric difference is \( Z - \{0\} \), which is all integers except \( 0 \).
**Hint**: Remember that symmetric difference includes elements that are in either set but not in both.
### Conclusion
After analyzing all options, we find that:
- Option 1 is incorrect.
- Option 2 is incorrect.
- Option 3 is correct.
- Option 4 is also correct.
Thus, the correct answer is that Option 3 and Option 4 are valid, but since the question seems to imply a single answer, we conclude with Option 3 being the most straightforward correct answer.
### Final Answer
**Option 3: \( Z - Z_N = Z_P \)** is correct.
