A full step by step solution please to the attached image hwk question. Both part a and b and c. 4. Let I denote the set of irrational numbers.(a). Prove that if r E I then Vn e Z*, nr E I.(b). Explain True or False of statement: “If S is an infinite subset of I, then S is uncountable."(c). Prove by contradiction: "I is uncountable."

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…

Answered: 4. Let I denote the set of irrational…

Elementary Geometry For College Students, 7e

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ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter2: Parallel Lines

Section2.CT: Test

Problem 3CT: To prove a theorem of the form "If P, then Q" by the indirect method, the first line of the proof...

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

4. Let I denote the set of irrational numbers.
(a). Prove that if r E I then Vn e Z*, nr E I.
(b). Explain True or False of statement: “If S is an infinite subset of I, then S is uncountable."
(c). Prove by contradiction: "I is uncountable."

Expert Solution

Step 1

PART A

Let $r \in I$ and $n \in Z^{+}$

Suppose $n r \notin I$ i.e. $n r \notin I$ i.e. $n r \in Q$ i.e. $n z$ is rational so,

$n r = \frac{a}{b}$ where $a, b \in Z$

Therefore, $r = \frac{a}{n b}$

Clearly, $a, b \in Z$ so $n b \in Z$ and $n b \neq 0$

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

Hence $n r$ is irrational.

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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.