Solve C, D, E please and Thankyou. ### Relations and Their PropertiesFor each of the following relations, determine whether the relation is:- **Reflexive**- **Anti-reflexive**- **Symmetric**- **Anti-symmetric**- **Transitive**- **A partial order**- **A strict order**- **An equivalence relation****Justify all your answers.**#### a. Relation on People**Description:** \( R \) is a relation on the set of all people such that \((a, b) \in R\) if and only if \(a\) and \(b\) have a common grandparent.#### b. Relation on Power Sets**Description:** \( R \) is a relation on the power set of a set \( A \) such that \((X, Y) \in R\) if and only if \( X \subseteq Y \).#### c. Relation on Integers \( \mathbb{Z} \)**Description:** \( R \) is a relation on \( \mathbb{Z} \) such that \((x, y) \in R\) if and only if \( xy \geq 0 \).#### d. Relation on Positive Integers \( \mathbb{Z}^+ \)**Description:** \( R \) is a relation on \( \mathbb{Z}^+ \) such that \((x, y) \in R\) if and only if there is a positive integer \( n \) such that \( x^n = y \).#### e. Relation on Real Numbers \( \mathbb{R} \)**Description:** \( R \) is a relation on \( \mathbb{R} \) such that \((x, y) \in R\) if and only if \( x - y \) is rational. **Hint:** The sum of two rational numbers is rational.

Answered: Image /qna-images/answer/45c68761-68db-4750-9e88-e6b0ef8f5e91.jpg

c. Ris a relation on Z such that (x, y) e R if and only if xy 2 0. d. Ris a relation on Z* such that (x, y) E R if and only if there is a positive integer n such that x" = y. e. Ris a relation on R such that (x, y) ER if and only if x – y is rational. Hint: The sum of two rational numbers is rational.

c. Ris a relation on Z such that (x, y) e R if and only if xy 2 0. d. Ris a relation on Z* such that (x, y) E R if and only if there is a positive integer n such that x" = y. e. Ris a relation on R such that (x, y) ER if and only if x – y is rational. Hint: The sum of two rational numbers is rational.

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

Solve C, D, E please and Thankyou.

### Relations and Their Properties

For each of the following relations, determine whether the relation is:

- **Reflexive**
- **Anti-reflexive**
- **Symmetric**
- **Anti-symmetric**
- **Transitive**
- **A partial order**
- **A strict order**
- **An equivalence relation**

**Justify all your answers.**

#### a. Relation on People
**Description:**
\( R \) is a relation on the set of all people such that \((a, b) \in R\) if and only if \(a\) and \(b\) have a common grandparent.

#### b. Relation on Power Sets
**Description:**
\( R \) is a relation on the power set of a set \( A \) such that \((X, Y) \in R\) if and only if \( X \subseteq Y \).

#### c. Relation on Integers \( \mathbb{Z} \)
**Description:**
\( R \) is a relation on \( \mathbb{Z} \) such that \((x, y) \in R\) if and only if \( xy \geq 0 \).

#### d. Relation on Positive Integers \( \mathbb{Z}^+ \)
**Description:**
\( R \) is a relation on \( \mathbb{Z}^+ \) such that \((x, y) \in R\) if and only if there is a positive integer \( n \) such that \( x^n = y \).

#### e. Relation on Real Numbers \( \mathbb{R} \)
**Description:**
\( R \) is a relation on \( \mathbb{R} \) such that \((x, y) \in R\) if and only if \( x - y \) is rational.
**Hint:** The sum of two rational numbers is rational.

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