To show that $R$ is an equivalence relation on the set of ordered pairs of positive integers, we need to verify that $R$ satisfies the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

1. Reflexivity: We need to show that for any $(a, b)$, $(a, b) R (a, b)$.

Consider $(a, b)$ where $a$ and $b$ are positive integers. We need to check if $(a, b) R (a, b)$. According to the definition of $R$, $(a, b) R (c, d)$ if and only if $ad = bc$. Here, $a = a$ and $b = b$, so we need to check if $ab = ba$.

Clearly, $ab = ba$ (since multiplication is commutative).

Thus, $(a, b) R (a, b)$ and $R$ is reflexive.

2. Symmetry: We need to show that if $(a, b) R (c, d)$, then $(c, d) R (a, b)$.

Assume $(a, b) R (c, d)$. By the definition of $R$, this means that $ad = bc$.

To show symmetry, we need to check if $cd = da$. Given that $ad = bc$, we can rearrange this equation to see that $cd = da$.

Thus, $(c, d) R (a, b)$ and $R$ is symmetric.

3. Transitivity: We need to show that if $(a, b) R (c, d)$ and $(c, d) R (e, f)$, then $(a, b) R (e, f)$.

Assume $(a, b) R (c, d)$ and $(c, d) R (e, f)$. By the definition of $R$, we have: $ad = bc \quad \text{and} \quad cf = de$

We need to show that $af = be$.

Starting from the given: $ad = bc \implies a = \frac{bc}{d}$ $cf = de \implies f = \frac{de}{c}$

Now, multiply $a$ and $f$: $af = \left(\frac{bc}{d}\right) \left(\frac{de}{c}\right) = \frac{bc \cdot de}{dc} = \frac{b \cdot de}{d} = be$

Thus, $af = be$ which implies $(a, b) R (e, f)$.

Therefore, $R$ is transitive.

Since $R$ is reflexive, symmetric, and transitive, $R$ is an equivalence relation on the set of ordered pairs of positive integers.

Would you like more details on any of these steps or have any questions?

Here are 8 related questions to expand on this topic:

1. Can you give an example of a set of ordered pairs that form an equivalence class under this relation?
2. How can we use the concept of equivalence classes to partition a set?
3. What are the general properties of equivalence relations?
4. How would you prove that a given relation is not an equivalence relation?
5. What are some real-world examples of equivalence relations?
6. How does the concept of an equivalence relation relate to modular arithmetic?
7. Can you describe the difference between a partial order and an equivalence relation?
8. How can equivalence relations be used in computer science, for example in data classification?

Tip: When working with equivalence relations, always verify reflexivity, symmetry, and transitivity to confirm that the relation holds across the entire set.