Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Concept explainers

Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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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**."
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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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