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The problem asks to select the collection of sets that forms a partition of \(\mathbb{Z}\), the set of all integers. 

A partition of a set divides it into disjoint subsets, meaning that every element of the set must belong to exactly one of the subsets, and the union of all the subsets must be the entire set.

Let's analyze each option:

### A: 
- \(\{x \in \mathbb{Z} : x < 0\}\)
- \(\{x \in \mathbb{Z} : x > 0\}\)
- \(\{0\}\)

This option includes all integers. The set of negative integers, the set of positive integers, and the single element \(\{0\}\) cover all integers. These sets are disjoint because no number can belong to more than one of these subsets. Hence, **this is a partition**.

### B:
- \(\{x \in \mathbb{Z} : x < 0\}\)
- \(\{x \in \mathbb{Z} : x > 0\}\)

This collection excludes \(\{0\}\), which means that the number 0 is missing, and it is not included in any set. Therefore, **this is not a partition**, as it doesn't cover all integers.

### C:
- \(\{x \in \mathbb{Z} : x \leq 0\}\)
- \(\{x \in \mathbb{Z} : x > 0\}\)
- \(\{0\}\)

In this case, \(\{x \in \mathbb{Z} : x \leq 0\}\) includes \(0\), but the third set is also \(\{0\}\), meaning \(0\) is repeated in two different sets. This violates the condition of a partition, as elements should be in only one set. Hence, **this is not a partition**.

### D:
- \(\{x \in \mathbb{Z} : x < 0\}\)
- \(\{x \in \mathbb{Z} : x \geq 0\}\)

This collection covers all integers, with one set for negative integers and one set for non-negative integers (including \(0\)). The sets are disjoint, and their union covers all integers. Therefore, **this is a partition**.

### Conclusion:
The correct collections of sets that form partitions are **A and D**.

Would you like further explanation or have any other questions?

Here are some related questions to explore:
1. What are the conditions for a partition of a set?
2. Can a partition have overlapping subsets?
3. What is the union of the sets in a partition?
4. Why is \(\{0\}\) necessary in some cases for partitioning \(\mathbb{Z}\)?
5. How does the inclusion of 0 affect partitioning integers?

**Tip:** When creating a partition of a set, always ensure that every element of the original set belongs to one and only one subset.

Select the collection of sets that forms a partition of ℤ.

Learn how to identify the collection of sets that forms a partition of the set of integers (ℤ). This problem covers set theory concepts, focusing on how to partition a set into disjoint subsets. Suitable for students in Grades 9-12.

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