Need help with this math problem. I received feedback from my Professor for this homework problem and he said Connex? I think he meant connex relation. Below the homework problem is my work. **1. Problem Statement**Let $ A $ be a nonempty set. Let $\sim$ be a relation on $\mathscr{P}(A)$ defined by:Letting $ B $ and $ C $ be arbitrary subsets of $ A $, $ B \sim C $ if and only if $ B \subseteq C $.What kind of order is $\sim$ (if any)? Prove the properties that hold, and construct counterexamples for the properties that don't.**1. Proof**$ A $ is given to be a non-empty set of 2 and $\sim$ is the relation on $\mathscr{P}(A)$ that is defined by:$ B $ and $ C $ be arbitrary subsets of $ A $, $ B \sim C $ if and only if $ B \subseteq C $.Since $ B \sim B $ because $ B \subseteq B $ (Every set is a subset of itself): \[\Rightarrow B \sim B \quad \forall \, B \in \mathscr{P}(A)\]Thus, $\sim$ is a Reflexive relation.Let $ B, C, $ and $ D $ be any three subsets of $ A $ such that $ B \subseteq C $ and $ C \subseteq B $.This implies $ B = C \Rightarrow \text{relation } (B \sim C \text{ if and only if } B \subseteq C) $.---Diagrams or graphs were not included in the image. A is given to be a non-empty set of 2 and ~ is the relation on ℘(A) that is defined byB and C be arbitrary subsets of A,B ~ C if and only if $ B \subseteq C $Since B ~ B ∵ $ B \subseteq B $ (Every set is a subset of itself)⇒ B ~ B ∀ B ∈ ℘(A)∴ ~ is a Reflexive relationLet B, C, and D be any three subsets of (A) of $ B \subseteq C $ and $ C \subseteq B $⇔ B = C ⇒ relation (B ~ C if and only if $ B \subseteq C $)~ is Anti-SymmetricNow let B ~ C ⇒ $ B \subseteq C $C ~ D ⇒ $ C \subseteq D $Since $ B \subseteq C $ and $ C \subseteq D $ ⇒ $ B \subseteq D $⇒ B ~ D⇒ ~ is TransitiveThus, relation '~' and partial ordering on ℘(A).

Answered: Image /qna-images/answer/de2958ed-c218-438d-b569-40d518930961.jpg

1. Let A be a nonempty set. Let ~ be a relation on (A) defined by: Letting B and C be arbitrary subsets of A, B-C iff BCC. What kind of order is~ (if any)? Prove the properties that hold, and construct counterexamples for the properties that don't.

1. Let A be a nonempty set. Let ~ be a relation on (A) defined by: Letting B and C be arbitrary subsets of A, B-C iff BCC. What kind of order is~ (if any)? Prove the properties that hold, and construct counterexamples for the properties that don't.

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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Question

Need help with this math problem. I received feedback from my Professor for this homework problem and he said Connex? I think he meant connex relation. Below the homework problem is my work.

**1. Problem Statement**

Let $ A $ be a nonempty set. Let $\sim$ be a relation on $\mathscr{P}(A)$ defined by:

Letting $ B $ and $ C $ be arbitrary subsets of $ A $, $ B \sim C $ if and only if $ B \subseteq C $.

What kind of order is $\sim$ (if any)? Prove the properties that hold, and construct counterexamples for the properties that don't.

**1. Proof**

$ A $ is given to be a non-empty set of 2 and $\sim$ is the relation on $\mathscr{P}(A)$ that is defined by:

$ B $ and $ C $ be arbitrary subsets of $ A $,
$ B \sim C $ if and only if $ B \subseteq C $.

Since $ B \sim B $ because $ B \subseteq B $ (Every set is a subset of itself):
\[
\Rightarrow B \sim B \quad \forall \, B \in \mathscr{P}(A)
\]
Thus, $\sim$ is a Reflexive relation.

Let $ B, C, $ and $ D $ be any three subsets of $ A $ such that $ B \subseteq C $ and $ C \subseteq B $.

This implies $ B = C \Rightarrow \text{relation } (B \sim C \text{ if and only if } B \subseteq C) $.

---

Diagrams or graphs were not included in the image.

A is given to be a non-empty set of 2 and ~ is the relation on ℘(A) that is defined by

B and C be arbitrary subsets of A,

B ~ C if and only if $ B \subseteq C $

Since B ~ B ∵ $ B \subseteq B $ (Every set is a subset of itself)

⇒ B ~ B ∀ B ∈ ℘(A)

∴ ~ is a Reflexive relation

Let B, C, and D be any three subsets of (A) of $ B \subseteq C $ and $ C \subseteq B $

⇔ B = C ⇒ relation (B ~ C if and only if $ B \subseteq C $)

~ is Anti-Symmetric

Now let B ~ C ⇒ $ B \subseteq C $

C ~ D ⇒ $ C \subseteq D $

Since $ B \subseteq C $ and $ C \subseteq D $ ⇒ $ B \subseteq D $

⇒ B ~ D

⇒ ~ is Transitive

Thus, relation '~' and partial ordering on ℘(A).

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