The image you provided contains detailed information regarding a relation, denoted by the tilde (~), being described as an equivalence relation. This is then followed by explanations, including an abstract definition based on integral domains and some discussion of rational numbers.

Key Concepts:

1. Definition of Equivalence Relation: A relation ~ is called an equivalence relation if it satisfies the following three properties:

   • Reflexive: $a \sim a$ for all $a \in A$.
   • Symmetric: If $a \sim b$, then $b \sim a$.
   • Transitive: If $a \sim b$ and $b \sim c$, then $a \sim c$.

2. Context from the Image:

   • Integral Domain: A subset $D \subseteq Z \times Z$ is discussed, where $Z$ represents integers. The relation $\sim$ is defined based on rational numbers $\frac{a_1}{b_1} = \frac{a_2}{b_2}$, meaning that for two pairs $(a_1, b_1)$ and $(a_2, b_2)$, they are related if and only if $a_1b_2 = a_2b_1$, which is a typical way of representing equivalence of fractions.

   • Equation (4.1): This seems to describe how pairs of integers are compared to verify equivalence, with conditions that $b_1$ and $b_2$ are non-zero, indicating that this relation holds between valid rational numbers.

3. Abstraction into an Integral Domain: The relation is applied to elements in the product set $Z \times Z$, and the discussion appears to abstract this into an integral domain where the properties of equivalence (reflexivity, symmetry, and transitivity) can be verified.

Proving Equivalence:

1. Reflexivity:

   • Any pair $(p, q) \in Z \times Z^*$ is related to itself because $\frac{p}{q} = \frac{p}{q}$, or equivalently $p \cdot q = p \cdot q$, which satisfies the condition.

2. Symmetry:

   • If $\frac{p_1}{q_1} = \frac{p_2}{q_2}$, then $p_1 q_2 = p_2 q_1$, and by symmetry, $p_2 q_1 = p_1 q_2$, hence the relation is symmetric.

3. Transitivity:

   • If $\frac{p_1}{q_1} = \frac{p_2}{q_2}$ and $\frac{p_2}{q_2} = \frac{p_3}{q_3}$, then $p_1 q_2 = p_2 q_1$ and $p_2 q_3 = p_3 q_2$. From this, you can deduce $p_1 q_3 = p_3 q_1$, hence transitivity holds.

Thus, the relation $\sim$ defined in the image is indeed an equivalence relation as it satisfies all three necessary conditions.

Would you like further details on any specific part of the proof or explanation? Here are some related questions you might find interesting:

1. What are the properties of an integral domain, and how do they relate to equivalence relations?
2. How is the concept of fractions formalized using pairs of integers in abstract algebra?
3. Can this definition of equivalence be extended to other mathematical structures beyond rational numbers?
4. What are other common examples of equivalence relations in mathematics?
5. How does this abstract definition apply to real-world problems involving proportional relationships?

Tip: When working with equivalence relations, always check the three fundamental properties: reflexivity, symmetry, and transitivity.