https://file.aiquickdraw.com/mathbot/file/1730342187177mn0z2zx3.jpg

Let's analyze each digraph to determine the properties for the relations defined by them. We need to assess whether each relation is:

1. **Reflexive**: Every element relates to itself. In graph terms, each vertex has a self-loop.
2. **Symmetric**: If there is an arrow from \(a\) to \(b\), there must also be an arrow from \(b\) to \(a\).
3. **Antisymmetric**: If there is an arrow from \(a\) to \(b\) and from \(b\) to \(a\), then \(a\) must equal \(b\).
4. **Transitive**: If there is an arrow from \(a\) to \(b\) and from \(b\) to \(c\), there should be an arrow from \(a\) to \(c\).

Let’s evaluate each digraph for these properties.

### Digraph Analysis:

#### (i)
- **Reflexive**: Yes, each vertex has a self-loop.
- **Symmetric**: No, as not all arrows are reciprocated (e.g., if there is an arrow from \(1\) to \(2\), there isn't necessarily one from \(2\) to \(1\)).
- **Antisymmetric**: No, because pairs like \(1 \rightarrow 2\) and \(2 \rightarrow 1\) exist.
- **Transitive**: Yes, as any reachable path implies direct connectivity.

#### (ii)
- **Reflexive**: No, not all vertices have self-loops.
- **Symmetric**: Yes, every connection is reciprocated.
- **Antisymmetric**: No, since reciprocal connections exist.
- **Transitive**: No, as there are paths without direct connections (i.e., missing some transitive connections).

#### (iii)
- **Reflexive**: No, not all vertices have self-loops.
- **Symmetric**: No, not all connections are reciprocated.
- **Antisymmetric**: Yes, no two different vertices have reciprocal connections.
- **Transitive**: No, because some indirect paths do not have direct connections.

#### (iv)
- **Reflexive**: Yes, each vertex has a self-loop.
- **Symmetric**: Yes, all connections are reciprocated.
- **Antisymmetric**: No, due to symmetric pairs.
- **Transitive**: Yes, as all possible paths are connected directly or reciprocally.

#### (v)
- **Reflexive**: No, not all vertices have self-loops.
- **Symmetric**: No, not all connections are reciprocated.
- **Antisymmetric**: Yes, reciprocal connections are absent for distinct vertices.
- **Transitive**: No, as some paths lack direct connectivity.

#### (vi)
- **Reflexive**: Yes, each vertex has a self-loop.
- **Symmetric**: No, as not all connections are reciprocated.
- **Antisymmetric**: Yes, since no reciprocal connections exist among distinct vertices.
- **Transitive**: Yes, as all paths satisfy the transitive property.

### Summary for Equivalence and Partial Orderings

- **Equivalence Relations** (Reflexive, Symmetric, Transitive): (iv)
- **Partial Orderings** (Reflexive, Antisymmetric, Transitive): (i) and (vi)

Would you like further details on any specific part of this analysis?

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### Relative Questions:

1. How would the interpretation change if one graph lacked transitivity but was symmetric?
2. What real-world scenarios could these relations model, especially in digraph form?
3. How does adding or removing self-loops affect reflexivity and equivalence?
4. Can a graph be antisymmetric without being reflexive? Why or why not?
5. Why are partial orders particularly useful in hierarchical systems?

### Tip:
In directed graphs, reflexivity is often identified by checking for self-loops on every vertex, making it a quick first property to evaluate.

Consider the relations defined by the digraphs in Figure 6.3.18. (a) Determine whether the given relations are reflexive, symmetric, antisymmetric, or transitive. Try to develop procedures for determining the validity of these properties from the graphs. (b) Which of the graphs are of equivalence relations or of partial orderings?

Explore how to determine if relations in directed graphs are reflexive, symmetric, antisymmetric, or transitive, and identify equivalence relations and partial orderings. This problem provides a detailed analysis of six digraphs and their properties, ideal for undergraduate students studying graph theory.

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