Let's break this down into three parts:

1. Writing $R$ as a set of ordered pairs:

Given the set $A = \{1, 2, 3, 4\}$ and the relation $R$, defined by "a divides b" (denoted as $a | b$), we can express the ordered pairs $(a, b)$ where $a$ divides $b$.

First, we check which elements of $A$ divide the others:

• $1 | 1$, $1 | 2$, $1 | 3$, $1 | 4$ (since 1 divides any number)
• $2 | 2$, $2 | 4$ (since 2 divides 2 and 4)
• $3 | 3$ (since 3 divides 3)
• $4 | 4$ (since 4 divides 4)

So, the relation $R$ as a set of ordered pairs is:

$R = \{(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)\}$

2. Drawing the directed graph of $R$:

• The graph will have four vertices: 1, 2, 3, and 4.
• Draw directed edges (arrows) from $a$ to $b$ whenever $a | b$.

The directed edges will be:

• $1 \to 1$, $1 \to 2$, $1 \to 3$, $1 \to 4$
• $2 \to 2$, $2 \to 4$
• $3 \to 3$
• $4 \to 4$

I'll explain how to visualize this graph further if needed.

3. Finding $R^{-1}$ (the inverse relation):

The inverse relation $R^{-1}$ consists of all pairs $(b, a)$ such that $(a, b)$ is in $R$. In other words, we reverse the ordered pairs of $R$.

For $R = \{(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)\}$, the inverse relation $R^{-1}$ is:

$R^{-1} = \{(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (3,3), (4,4)\}$

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Do you have any questions or want further details?

Here are 5 related questions to deepen your understanding:

1. How does one determine if a relation is reflexive or symmetric based on ordered pairs?
2. Can a relation be symmetric if $a$ divides $b$?
3. What are the properties of the relation $R$ in terms of reflexivity, symmetry, and transitivity?
4. How does the directed graph of a relation help in visualizing its properties?
5. Can you apply the concept of inverse relations to more complex sets and relations?

Tip: When drawing directed graphs, always start by identifying self-loops (like $a \to a$) to check for reflexive elements!