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.