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.