Let A = {1, 2, 3, 4}, and R is a relation defined by “a divides b:. Write R as a set of ordered pair, draw

directed graph. Also find R-1

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!

Let A = {1, 2, 3, 4}, and R is a relation defined by 'a divides b'. Write R as a set of ordered pairs, draw directed graph, and find R-1.

Explore the relation R on the set A = {1, 2, 3, 4} where a divides b. Learn how to express R as ordered pairs, draw the directed graph, and find the inverse relation R-1. This problem covers concepts in set theory, relations, and inverse relations, ideal for Grades 9-12.

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