Which of the following is incorrect?

Options

A. Positive integer `gt` zero `gt` negative integer
B. Positive integer `gt` negative integer `lt` zero
C. Zero `lt` positive integer `gt` negative integer
D. Positive integer `gt` zero `lt` negative integer

Answer

To determine which of the given options is incorrect regarding integers, we will analyze each option based on the properties of positive integers, negative integers, and zero.

### Step-by-Step Solution:

1. **Understanding the Number Line**:
- The number line is divided into three main parts: negative integers (to the left of zero), zero (at the center), and positive integers (to the right of zero).
- Negative integers are less than zero, and positive integers are greater than zero.

**Hint**: Visualize the number line to understand the positions of negative integers, zero, and positive integers.

2. **Analyzing Each Option**:
- **Option A**: Positive integer > 0 > Negative integer
- This statement is true because positive integers are indeed greater than zero, and zero is greater than negative integers.

**Hint**: Remember that positive integers start from 1 and go upwards, while negative integers start from -1 and go downwards.

- **Option B**: Positive integers < Negative integers < 0
- This statement is incorrect. Positive integers are not less than negative integers; they are greater than zero and thus greater than negative integers.

**Hint**: Compare the values: positive integers are always to the right of zero on the number line.

- **Option C**: 0 < Positive integer
- This statement is true because zero is less than any positive integer.

**Hint**: Any positive integer (like 1, 2, 3, etc.) is always greater than zero.

- **Option D**: 0 < Negative integer
- This statement is incorrect. Zero is not less than negative integers; it is greater than all negative integers.

**Hint**: Remember that negative integers are always to the left of zero on the number line.

3. **Conclusion**:
- The incorrect options are B and D. However, since we are looking for one incorrect option, we can conclude that both B and D are incorrect, but if we have to pick one based on the analysis, we can choose D as the more explicit incorrect statement regarding zero and negative integers.

### Final Answer:
The incorrect option is **D**: 0 < Negative integer.