Question

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

1) {Z}
2) {1, 2, 3, ...}
3) {2, 4, 6, ...}
4) {odd numbers}

Answer 1

None of the provided options form a partition of the set of integers Z on their own. To form a partition, the sets need to be non-overlapping and cover all integers completely, which is not achieved by any single option given.

Step-by-step explanation:

The student has asked which collection of sets forms a partition of the set of integers Z. A partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in exactly one subset. For a collection of sets to form a partition of Z, the sets need to be non-overlapping and collectively exhaustive. Looking at the options:

• {Z} is the set of all integers and does not form a partition on its own.
• {1, 2, 3, ...} represents the set of all natural numbers and does not include negative integers or zero.
• {2, 4, 6, ...} represents the set of all positive even numbers and omits odd integers and all negative integers.
• {odd numbers} is the set of all odd integers but does not include even integers.

None of the provided options individually represent a partition of Z, as they do not satisfy the requirements of being non-overlapping and covering all integers. A proper partition of Z, for example, would be the sets of positive integers, negative integers, and {0}. This would ensure that every integer is included in exactly one subset.