Let's start by reviewing the details we have to create the arrow diagram for the relation $R$.

Problem Summary:

• We have sets $A = \{2, 3, 4\}$ and $B = \{6, 8, 10\}$.
• Relation $R$ is defined such that for any $(x, y)$ in $A \times B$, the relation $(x, y) \in R$ holds if $\frac{y}{x}$ is an integer.

Steps:

1. Identify all valid pairs $(x, y)$ such that $\frac{y}{x}$ is an integer.

   • For $x = 2$:
     • $\frac{6}{2} = 3$ (integer), so $(2, 6) \in R$
     • $\frac{8}{2} = 4$ (integer), so $(2, 8) \in R$
     • $\frac{10}{2} = 5$ (integer), so $(2, 10) \in R$
   • For $x = 3$:
     • $\frac{6}{3} = 2$ (integer), so $(3, 6) \in R$
     • $\frac{8}{3} \neq \text{integer}$, so $(3, 8) \notin R$
     • $\frac{10}{3} \neq \text{integer}$, so $(3, 10) \notin R$
   • For $x = 4$:
     • $\frac{6}{4} \neq \text{integer}$, so $(4, 6) \notin R$
     • $\frac{8}{4} = 2$ (integer), so $(4, 8) \in R$
     • $\frac{10}{4} \neq \text{integer}$, so $(4, 10) \notin R$

2. List the ordered pairs in the relation $R$:

   • The ordered pairs are: $R = \{(2, 6), (2, 8), (2, 10), (3, 6), (4, 8)\}$

Arrow Diagram:

Now, to draw the arrow diagram, we represent each element of $A$ and $B$ as points. For each pair in $R$, we draw an arrow from the element in $A$ to the corresponding element in $B$.

• $2 \to 6, 2 \to 8, 2 \to 10$
• $3 \to 6$
• $4 \to 8$

This is how the relations from $A$ to $B$ should be diagrammatically represented. I'll now summarize how the diagram would appear:

• $A = \{2, 3, 4\}$ are represented on the left.
• $B = \{6, 8, 10\}$ are represented on the right.
• Arrows are drawn from:
  • $2$ to $6$, $8$, and $10$
  • $3$ to $6$
  • $4$ to $8$

Would you like me to generate a visual of this diagram for you?

Here are some follow-up questions to deepen your understanding:

1. Can you describe the process to verify if any arbitrary pair $(x, y)$ belongs to $R$?
2. How would the arrow diagram change if $B$ had different values, say $B = \{12, 14, 15\}$?
3. What is the significance of the condition $\frac{y}{x}$ being an integer in the context of relations?
4. Can you think of another relation with a different condition (e.g., $x + y$ is even) and describe its arrow diagram?
5. How would the co-domain of $R$ change if we added new elements to $B$?

Tip: For relations, always check each pair carefully to ensure that the condition defining the relation is satisfied!