Let's break down and solve the problem step by step. We are given that:

• $R$ is a relation from set $A$ to set $B$, and the relation is defined as $R = \{(a, b) \mid a \text{ divides } b\}$, where $a, b$ are positive integers.
• We are asked to find the inverse relation $R^{-1}$ and the complementary relation $R^c$.

Part (a) – Find $R^{-1}$

The inverse of a relation $R$, denoted $R^{-1}$, consists of the ordered pairs $(b, a)$ such that $(a, b) \in R$. In other words, if $a$ divides $b$ in $R$, then in $R^{-1}$, $b$ divides $a$.

Thus, if $(a, b) \in R$, then $a$ divides $b$. Therefore, for $(b, a) \in R^{-1}$, we must have that $b$ divides $a$.

So, the relation $R^{-1}$ is:

$R^{-1} = \{(b, a) \mid b \text{ divides } a\}$

Part (b) – Find $R^c$

The complementary relation $R^c$ consists of all the ordered pairs $(a, b)$ such that $(a, b) \notin R$. In other words, $R^c$ contains all pairs where $a$ does not divide $b$.

Thus, the complementary relation $R^c$ is:

$R^c = \{(a, b) \mid a \text{ does not divide } b\}$

Summary of Answers:

• (a) The inverse relation $R^{-1}$ is:

$R^{-1} = \{(b, a) \mid b \text{ divides } a\}$

• (b) The complementary relation $R^c$ is:

$R^c = \{(a, b) \mid a \text{ does not divide } b\}$

Would you like more detailed explanations or examples of these relations?

Here are some related questions to expand on this topic:

1. What is the difference between a relation and a function, and how does it apply to divisibility relations?
2. How can we prove whether a relation is reflexive, symmetric, or transitive?
3. How do we find the composition of relations, particularly with divisibility relations?
4. What is the concept of an equivalence relation, and does the divisibility relation satisfy it?
5. How do we represent a relation as a matrix and analyze its properties?

Tip: When working with relations, always carefully consider the properties (reflexivity, symmetry, transitivity) that can help in understanding and solving problems related to the relation.