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Let P = {X E P(Z*)|X is finite}. Prove that P is denumerable.

Let P = {X E P(Z*)|X is finite}. Prove that P is denumerable.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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## Problem Statement **4.** Let \( P = \{ X \in \mathcal{P}(\mathbb{Z}^+) \mid X \text{ is finite} \} \). Prove that \( P \) is denumerable. ### Explanation Here, \( \mathcal{P}(\mathbb{Z}^+) \) denotes the power set of the positive integers \( \mathbb{Z}^+ \), that is, the set of all subsets of \( \mathbb{Z}^+ \). The set \( P \) consists of all finite subsets of \( \mathbb{Z}^+ \). ### Objective The task is to prove that the set \( P \), which contains all finite subsets of \( \mathbb{Z}^+ \), is denumerable. This means showing that there exists a one-to-one correspondence between the elements of \( P \) and the set of positive integers, i.e., \( P \) can be listed in a sequence indexed by positive integers.
Transcribed Image Text:## Problem Statement **4.** Let \( P = \{ X \in \mathcal{P}(\mathbb{Z}^+) \mid X \text{ is finite} \} \). Prove that \( P \) is denumerable. ### Explanation Here, \( \mathcal{P}(\mathbb{Z}^+) \) denotes the power set of the positive integers \( \mathbb{Z}^+ \), that is, the set of all subsets of \( \mathbb{Z}^+ \). The set \( P \) consists of all finite subsets of \( \mathbb{Z}^+ \). ### Objective The task is to prove that the set \( P \), which contains all finite subsets of \( \mathbb{Z}^+ \), is denumerable. This means showing that there exists a one-to-one correspondence between the elements of \( P \) and the set of positive integers, i.e., \( P \) can be listed in a sequence indexed by positive integers.
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