https://file.aiquickdraw.com/mathbot/file/1730247642688sghs5abc.jpg

To answer this question, let's examine the directed graph provided and translate each arrow into an ordered pair. Each directed edge \( (a \rightarrow b) \) represents a relation \( (a, b) \) between two elements.

### Analysis of the Directed Graph
1. **Loop at node 1**: This indicates the pair \( (1, 1) \).
2. **Edge from 1 to 2**: This gives the pair \( (1, 2) \).
3. **Edge from 2 to 4**: This results in the pair \( (2, 4) \).
4. **Edge from 3 to 1**: This results in the pair \( (3, 1) \).
5. **Edge from 3 to 4**: This gives the pair \( (3, 4) \).
6. **Loop at node 4**: This indicates the pair \( (4, 4) \).

### Constructing the Set of Ordered Pairs
The relation as a set of ordered pairs is:
\[
\{ (1, 1), (1, 2), (2, 4), (3, 1), (3, 4), (4, 4) \}
\]

### Matching the Answer Options
Looking at the provided options, the set that matches our findings is:
\[
\{ (1, 1), (1, 2), (2, 4), (3, 1), (3, 4), (4, 4) \}
\]
This corresponds to the third option in the list.

### Final Answer
The correct answer is:
**Option 3: \( \{ (1, 1), (1, 2), (2, 4), (3, 1), (3, 4), (4, 4) \} \)**

Would you like more details or have any questions?

### Related Questions to Expand Understanding
1. How would the answer change if additional loops were added to nodes 2 or 3?
2. What properties does this relation have (e.g., reflexive, symmetric, transitive)?
3. How would you describe a directed graph that represents a symmetric relation?
4. What is the significance of loops in directed graphs in terms of relations?
5. How would you represent this relation as a matrix?

### Tip
In directed graphs, loops represent elements that relate to themselves, which is essential for understanding reflexive properties in relations.

Below is a directed graph on the set {1, 2, 3, 4}. Write the relation as a set of ordered pairs.

Learn how to determine the relation as a set of ordered pairs from a directed graph on the set {1, 2, 3, 4}. This problem involves interpreting directed edges as ordered pairs and constructing the relation from a given graph structure. Suitable for college-level students studying set theory and relations.

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