
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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A full step by step solution please to the attached image hwk question. Both part a and b and c.

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

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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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