Question

The sets 2n - this is the set of all even integers and n ∈ ℤ - this is the set of all odd integers form a partition.

a) Define and explain the concept of a partition in the context of sets and integers.
b) Instruct the respondent to consider the provided sets of even and odd integers and evaluate whether they form a partition.
c) Encourage a brief explanation or justification for the chosen answer, demonstrating an understanding of the characteristics that define a partition.

Ensure clarity in the question to prompt a thoughtful assessment of whether the given sets constitute a partition.

Answer 1

A partition divides a set into comprehensive, non-overlapping subsets. The sets of even and odd integers form a valid partition of the integers, fulfilling the requirements of being disjoint and covering the entire set.

Step-by-step explanation:

Understanding Partitions in Set Theory

A partition of a set is a division of the set's elements into non-empty subsets, in such a way that every element is included in exactly one subset. This means that the subsets cover the entire original set without overlapping. In the context of integers, the sets 2n and 2n + 1 , representing even and odd integers respectively, form a partition because:

• Every integer is either even or odd, so they cover the entire set of integers (comprehensive).
• No integer can be both even and odd, ensuring the subsets are disjoint (no overlap).
• Each subset is non-empty as there are infinitely many even and odd integers.

Considering these characteristics, the given sets of even and odd integers do indeed form a partition of the integers.