Help me understand these two problems. Problem 5A problem about inductive or dense sets like that, if both A and B are inductive, then areAnB and AU B also inductive? If both A and B are dense, then are AnB and AUB also dense? Why? Problem 2: Important theorems and results such as1. Principle of Mathematical Induction (Page 6)2. Completeness Axiom (Page 8)3. Theorem 1.5 Archimedean Property (Page 13)4. Theorem 1.9 Density of rational numbers (Page 15)5. Corollary 1.10 Density of irrational numbers (Page 15)6. Theorem 1.11 Triangle Inequality (Page 17)

Answered: A problem about inductive or dense sets…

A problem about inductive or dense sets like that, if both A and B are inductive, then are An B and AUB also inductive? If both A and B are dense, then are An B and AUB also dense? Why? Problem 5

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter8: Complex Numbers And Polarcoordinates

Section: Chapter Questions

Problem 2RP: A Bitter Dispute With the publication of Ars Magna, a dispute intensified between Jerome Cardan and...

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Question

Help me understand these two problems.

Problem 5
A problem about inductive or dense sets like that, if both A and B are inductive, then are
AnB and AU B also inductive? If both A and B are dense, then are AnB and AUB also dense? Why?

Problem 2
: Important theorems and results such as
1. Principle of Mathematical Induction (Page 6)
2. Completeness Axiom (Page 8)
3. Theorem 1.5 Archimedean Property (Page 13)
4. Theorem 1.9 Density of rational numbers (Page 15)
5. Corollary 1.10 Density of irrational numbers (Page 15)
6. Theorem 1.11 Triangle Inequality (Page 17)

Expert Solution

Step 1

Since you have asked multiple questions, we will solve the first Question for you. If you want any specific question to be solved, then please specify the question number or post only that question.

Given that the set $A$ and $B$ are inductive sets.

We know that a set $X$ is an inductive set if:

$ϕ \in X$
$x \in X$ implies that $x \cup \{x\} \in X$ .

(a).

Consider the set $A \cup B$ .

Since $A$ and $B$ are inductive sets, therefore:

$ϕ \in A$ and $ϕ \in B$ , therefore $ϕ \in A \cup B$ .

Suppose $x \in A \cup B$ , therefore either $x \in A$ or $x \in B$ .

Therefore without loss of generality let $x \in A$ .

Since $A$ is an inductive set, therefore $x \cup \{x\} \in A$ and hence $x \cup \{x\} \in A \cup B$ .

Hence $A \cup B$ is an inductive set.

Consider the set $A \cap B$ .

Since $A$ and $B$ are inductive sets, therefore:

$ϕ \in A$ and $ϕ \in B$ , therefore $ϕ \in A \cap B$ .

Suppose $x \in A \cap B$ , therefore $x \in A$ and $x \in B$ .

Since $A$ is an inductive set and $x \in A$ , therefore $x \cup \{x\} \in A$ and hence $x \cup \{x\} \in A \cap B$ .

Hence $A \cap B$ is an inductive set.

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