To determine whether the relations represented by the directed graphs in exercises 21–23 are equivalence relations, we need to check if each relation satisfies three properties of an equivalence relation:

1. Reflexivity: Every node should have a loop (an edge from the node to itself).
2. Symmetry: If there is a directed edge from node $x$ to node $y$, there should also be a directed edge from $y$ to $x$.
3. Transitivity: If there is a directed edge from node $x$ to node $y$, and from node $y$ to node $z$, there must also be a directed edge from $x$ to $z$.

Let's analyze each graph:

21.

• Reflexivity: Every node ($a$, $b$, $c$, $d$) has a loop, so it is reflexive.
• Symmetry: There is an edge from $a$ to $c$ and from $a$ to $d$, but no reverse edges from $c$ to $a$ or $d$ to $a$, so it is not symmetric.
• Transitivity: Since it's not symmetric, we don't need to check for transitivity in this case.

This graph is not an equivalence relation.

22.

• Reflexivity: Every node ($a$, $b$, $c$, $d$) has a loop, so it is reflexive.
• Symmetry: There is an edge from $a$ to $d$, and from $b$ to $c$, but the reverse edges are missing, so it is not symmetric.
• Transitivity: Again, since it's not symmetric, we don't need to check for transitivity.

This graph is not an equivalence relation.

23.

• Reflexivity: Every node ($a$, $b$, $c$, $d$) has a loop, so it is reflexive.
• Symmetry: For every edge from $x$ to $y$, there is a reverse edge from $y$ to $x$. Hence, it is symmetric.
• Transitivity: For any pair of edges, the transitivity condition holds. For example, since there is an edge from $a$ to $b$ and from $b$ to $c$, there is also an edge from $a$ to $c$.

This graph is an equivalence relation.

Would you like further details or have any questions?

Here are 5 related questions:

1. What are some real-life examples of equivalence relations?
2. How does the property of transitivity apply to directed graphs?
3. Can a relation be reflexive and symmetric but not transitive?
4. How would the graph change if it lost reflexivity but retained the other properties?
5. How does the concept of equivalence classes arise from equivalence relations?

Tip: Always verify reflexivity, symmetry, and transitivity sequentially when determining whether a relation is an equivalence relation.