https://file.aiquickdraw.com/mathbot/file/1728246430659ld2mbhhn.jpg

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?

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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!

Let A = {2, 3, 4} and B = {6, 8, 10} and define a relation R from A to B as follows: For all (x, y) ∈ A × B, (x, y) ∈ R means that y/x is an integer. Answer the following:
 a. Is 4R6? Is 4R8? Is (3,8) ∈ R? Is (2,10) ∈ R?
 b. Write R as a set of ordered pairs.
 c. Write the domain and co-domain of R.
 d. Draw an arrow diagram for R.

Learn how to define and illustrate a relation R from sets A = {2, 3, 4} to B = {6, 8, 10} using the condition y/x is an integer. This detailed explanation includes verifying ordered pairs, writing the relation as a set of ordered pairs, defining the domain and co-domain, and creating an arrow diagram. Suitable for high school students (Grades 10-12).

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