Let's analyze each part of this problem with respect to the relations $R_1$ and $R_2$ defined on the set of all positive integers:

• $R_1 = \{(a, b) \mid a \text{ divides } b\}$
• $R_2 = \{(a, b) \mid a \text{ is a multiple of } b\}$

For each part, we need to determine specific operations on these relations:

a) $R_1 \cup R_2$

The union $R_1 \cup R_2$ represents the set of pairs $(a, b)$ such that either $a$ divides $b$ or $a$ is a multiple of $b$ (or both). This includes all pairs where there is any divisibility relationship in either direction between $a$ and $b$.

b) $R_1 \cap R_2$

The intersection $R_1 \cap R_2$ represents the set of pairs $(a, b)$ such that $a$ divides $b$ and $a$ is also a multiple of $b$. This condition is satisfied if and only if $a = b$, because for any two distinct integers $a$ and $b$, one cannot simultaneously divide the other and be a multiple of it.

Thus, $R_1 \cap R_2 = \{(a, a) \mid a \in \mathbb{Z}^+ \}$, which is the set of all pairs where $a = b$.

c) $R_1 - R_2$

The difference $R_1 - R_2$ represents the set of pairs $(a, b)$ such that $a$ divides $b$ but $a$ is not a multiple of $b$. This includes pairs where $a \neq b$ and $a$ divides $b$.

d) $R_2 - R_1$

The difference $R_2 - R_1$ represents the set of pairs $(a, b)$ such that $a$ is a multiple of $b$ but $a$ does not divide $b$. This case is a bit unusual because if $a$ is a multiple of $b$, $a$ typically divides $b$. Therefore, for positive integers, this difference would generally yield an empty set, as it’s not possible to satisfy the conditions of $R_2$ without also satisfying $R_1$.

e) $R_1 \oplus R_2$

The symmetric difference $R_1 \oplus R_2$ represents the set of pairs $(a, b)$ that are in either $R_1$ or $R_2$, but not in both. This would include:

• Pairs where $a$ divides $b$ but $a$ is not a multiple of $b$ (from $R_1 - R_2$)
• Pairs where $a$ is a multiple of $b$ but $a$ does not divide $b$ (from $R_2 - R_1$), though, as discussed in part d, this is likely an empty set for positive integers.

Thus, $R_1 \oplus R_2$ primarily consists of pairs where $a \neq b$ and $a$ divides $b$.

Would you like further details on any of these answers? Here are some questions you may find useful to explore:

1. Can we list specific examples for each part to better understand these relations?
2. What are the properties of the relations $R_1$ and $R_2$ in terms of reflexivity, symmetry, and transitivity?
3. How would these relations change if defined on a different set, such as integers instead of positive integers?
4. How does the concept of symmetric difference apply in other types of mathematical relations?
5. Is there an efficient way to verify each condition computationally?

Tip: Understanding the properties of divisibility and multiples can often simplify complex problems involving relations on integers.