Let's analyze the directed graph and determine whether the relation it represents is:

Part a) An equivalence relation

For a relation to be an equivalence relation, it must satisfy the following three properties:

1. Reflexivity: Every element must have a loop to itself, meaning $(x, x)$ must be in the relation for all $x$.
2. Symmetry: If $(x, y)$ is in the relation, then $(y, x)$ must also be in the relation.
3. Transitivity: If $(x, y)$ and $(y, z)$ are in the relation, then $(x, z)$ must also be in the relation.

From the graph:

• Reflexivity: Each of the nodes $a$, $b$, $c$, and $d$ has a loop to itself, so the relation is reflexive.

• Symmetry: For symmetry, we need to check if for every arrow going from one node to another, there is a reverse arrow. For example, there is an arrow from $a \to d$, but no arrow from $d \to a$. Therefore, the relation is not symmetric.

• Transitivity: We need to check if for any pair of arrows $(x, y)$ and $(y, z)$, there is an arrow $(x, z)$. In this case, transitivity is hard to verify without specific paths leading through three nodes, but since the relation already fails symmetry, it doesn't matter for equivalence.

Thus, the relation is not an equivalence relation because it fails the symmetry condition.

Part b) A partial ordering

For a relation to be a partial order, it must satisfy:

1. Reflexivity: As mentioned earlier, the graph is reflexive.
2. Antisymmetry: If $(x, y)$ and $(y, x)$ are both in the relation, then $x = y$.
3. Transitivity: If $(x, y)$ and $(y, z)$ are in the relation, then $(x, z)$ must also be in the relation.

From the graph:

• Reflexivity is satisfied since every node has a self-loop.
• Antisymmetry: There is an arrow from $a \to d$, but no arrow from $d \to a$, and similarly no opposite pairs exist for the other directed edges. Thus, antisymmetry is satisfied.
• Transitivity: We need to check the transitivity condition, which appears to hold based on the structure, as there are no conflicting transitive paths.

Thus, the relation is a partial order because it satisfies reflexivity, antisymmetry, and transitivity.

Let me know if you want further details or have any questions!

Here are 5 related questions to deepen your understanding:

1. How does the symmetry property differ between equivalence relations and partial orders?
2. Can a relation be both an equivalence relation and a partial order? Why or why not?
3. What would be the impact on the relation if there were an edge from $d \to a$?
4. How can we visually identify a transitive relation in a directed graph?
5. What changes would need to be made to this graph for it to represent a total order?

Tip: For partial orders, a helpful concept is the Hasse diagram, which simplifies visualizing the structure by removing reflexive loops and implied transitive edges.