https://file.aiquickdraw.com/mathbot/file/1726472146200ldwato1o.jpg

I see you've uploaded an assignment with several questions regarding relations and equivalence relations. Let's go through each question step by step:

---

### 1. **Given \( A = \{1, 2, 3, 4\} \), give an example of a relation on \( A \) that is:**
   - **a. Reflexive and symmetric but not transitive.**
   - **b. Symmetric and transitive but not reflexive.**
   - **c. Transitive but neither reflexive nor symmetric.**

For each subquestion, you need to construct a set of ordered pairs that fulfill the required conditions (e.g., reflexive, symmetric, transitive).

---

### 2. **A relation \( \mathcal{R} \) is defined on \( \mathbb{Z} \) by \( x \mathcal{R} y \) if \( 3x - 7y \) is even. Prove that \( \mathcal{R} \) is an equivalence relation.**

To prove this, we need to show:
- Reflexivity: For all \( x \in \mathbb{Z} \), \( x \mathcal{R} x \).
- Symmetry: If \( x \mathcal{R} y \), then \( y \mathcal{R} x \).
- Transitivity: If \( x \mathcal{R} y \) and \( y \mathcal{R} z \), then \( x \mathcal{R} z \).

---

### 3. **Suppose that \( A = \{ (x, y) \in \mathbb{Z} \times \mathbb{Z} : y \neq 0 \} \) and define a relation \( \mathcal{R} \) on \( A \) by \( (x, y) \mathcal{R} (s, t) \) if and only if \( x t = y s \). Show that \( \mathcal{R} \) is an equivalence relation on \( A \).**

You must show that this relation satisfies reflexivity, symmetry, and transitivity based on the given condition \( xt = ys \).

---

### 4. **Prove or disprove: The union of two equivalence relations on a nonempty set is an equivalence relation.**

You need to analyze whether the union of two equivalence relations preserves the properties of reflexivity, symmetry, and transitivity.

---

### 5. **Prove or disprove: The intersection of two equivalence relations on a nonempty set is an equivalence relation.**

Here, you must investigate whether the intersection of two equivalence relations maintains reflexivity, symmetry, and transitivity.

---

Would you like to go through a specific question in more detail, or explore an example for any of them?

Here are some questions related to the topic:
1. Can an equivalence relation be non-transitive in some cases?
2. Why does the condition of symmetry not necessarily imply transitivity in relations?
3. How do you construct a non-reflexive but symmetric relation on a finite set?
4. How can we visualize relations using directed graphs?
5. How does the Cartesian product \( \mathbb{Z} \times \mathbb{Z} \) relate to equivalence relations?

**Tip:** When working with relations, always check if each property (reflexive, symmetric, transitive) is independent from the others—it's possible for a relation to satisfy one property but fail another.

Let A = {1, 2, 3, 4}. Give an example of a relation on A that is: a) reflexive and symmetric but not transitive; b) symmetric and transitive but not reflexive; c) transitive but neither reflexive nor symmetric. Other questions relate to equivalence relations and properties such as union and intersection of equivalence relations.

This problem set covers fundamental concepts in set theory and relations. It includes examples and proofs of reflexive, symmetric, and transitive relations. Also, explore the properties of equivalence relations, including their union and intersection. Suitable for undergraduate mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation