To determine if the relation in the directed graph is an equivalence relation, we need to analyze whether the relation meets the three properties required for an equivalence relation: reflexivity, symmetry, and transitivity.

1. Reflexivity: Each vertex must have a loop, meaning there must be an edge from each vertex to itself. In this graph, we can see that vertices $a$, $b$, and $c$ have loops, but $d$ does not. Therefore, the relation is not reflexive because not every vertex has an edge back to itself.

2. Symmetry: For each directed edge $(x, y)$, there should be an antiparallel edge $(y, x)$. Observing the graph, it appears that some pairs do have antiparallel edges, but not all pairs meet this criterion. This lack of symmetry means the relation does not satisfy the symmetry property either.

3. Transitivity: If there exists an edge from $x$ to $y$ and from $y$ to $z$, there should also be an edge from $x$ to $z$. Given the graph structure, this property is not uniformly satisfied across all vertices and edges.

Since the graph fails to meet the requirements for reflexivity, symmetry, and transitivity, the relation in this directed graph is not an equivalence relation.

Answer: False.

Would you like further details or have any questions?

Here are some related questions that might deepen your understanding:

1. What are examples of equivalence relations in mathematics or real-world contexts?
2. How can a directed graph be modified to satisfy reflexivity, symmetry, and transitivity?
3. Can a graph with missing reflexive loops ever satisfy all the conditions of an equivalence relation?
4. How is the concept of an antiparallel edge used in graph theory?
5. What are some common applications of equivalence relations in computer science and data organization?

Tip: When analyzing relations in directed graphs, always check for self-loops for reflexivity and matching antiparallel edges for symmetry.