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4. Let I denote the set of irrational numbers. (a). Prove that if r e I then V n E Z*, nr € I.

4. Let I denote the set of irrational numbers. (a). Prove that if r e I then V n E Z*, nr € I.

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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A full step by step solution please to the attached image hwk question. Both part a and b and c.

## Problem 4 Let \( I \) denote the set of **irrational numbers**. ### Part (a) **Prove** that if \( r \in I \) then \( \forall n \in \mathbb{Z}^+, nr \in I \). ### Part (b) **Explain** True or False for the statement: "If \( S \) is an **infinite subset** of \( I \), then \( S \) is **uncountable**." ### Part (c) **Prove by contradiction**: " \( I \) is **uncountable**."
Transcribed Image Text:## Problem 4 Let \( I \) denote the set of **irrational numbers**. ### Part (a) **Prove** that if \( r \in I \) then \( \forall n \in \mathbb{Z}^+, nr \in I \). ### Part (b) **Explain** True or False for the statement: "If \( S \) is an **infinite subset** of \( I \), then \( S \) is **uncountable**." ### Part (c) **Prove by contradiction**: " \( I \) is **uncountable**."
Expert Solution
Step 1

PART A

Let r∈I and n∈Z+

Suppose nr∉I i.e. nr∉I i.e. nr∈Q i.e. nz is rational so, 

nr=ab where a,b∈Z

Therefore,  r=anb

Clearly, a,b∈Z so nb∈Z and nb≠0

So r is rational which contradict the fact that r is irrational.

Hence nr is irrational.

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