Let's analyze the requirements for Question 3 based on the image you provided:

1. Definition of Graph: A graph is a mathematical structure used to model pairwise relations between objects. It consists of vertices (or nodes) and edges (which can be directed or undirected) that connect these vertices.

2. Ordered Pairs in the Relation: The directed graph in the image contains vertices labeled 1, 2, 3, and 4. Directed edges represent relations, so we list the ordered pairs based on the arrows' directions:

   • Observing the arrows in the graph:
     • From vertex 1: $(1, 2), (1, 3), (1, 4)$
     • From vertex 2: $(2, 1), (2, 3)$
     • From vertex 3: $(3, 1), (3, 2), (3, 4)$
     • From vertex 4: $(4, 1), (4, 2)$

   So, the set of ordered pairs representing this directed relation is: $\{ (1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2) \}$

3. Properties of the Relation:

   • Reflexive: A relation is reflexive if every element is related to itself, meaning for all $x$, $(x, x)$ should be present. In this graph, there are no loops (self-relations) for any vertices, so the relation is not reflexive.

   • Symmetric: A relation is symmetric if for every $(a, b)$, $(b, a)$ also exists. For instance, if $(1, 2)$ exists, then $(2, 1)$ must also exist for symmetry. Observing the pairs, we can see that not all pairs have their reverse included. Therefore, the relation is not symmetric.

   • Antisymmetric: A relation is antisymmetric if for every pair $(a, b)$ and $(b, a)$, $a$ must be equal to $b$. Since we have pairs like $(1, 2)$ and $(2, 1)$ with $1 \neq 2$, the relation is not antisymmetric.

   • Transitive: A relation is transitive if for every $(a, b)$ and $(b, c)$, $(a, c)$ must also exist. Examining the pairs, transitivity is violated in some cases (for instance, $(1, 2)$ and $(2, 3)$ are present, but $(1, 3)$ is not consistently resulting). Thus, the relation is not transitive.

4. Partial Order or Equivalence Relation:

   • A partial order is a relation that is reflexive, antisymmetric, and transitive. Since this relation is not reflexive or antisymmetric, it cannot be a partial order.
   • An equivalence relation requires the relation to be reflexive, symmetric, and transitive. As this relation does not meet any of these properties, it is not an equivalence relation.

Summary:

• Ordered Pairs: $\{ (1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2) \}$
• Properties:
  • Reflexive: No
  • Symmetric: No
  • Antisymmetric: No
  • Transitive: No
• Type of Relation: Neither a partial order nor an equivalence relation.

Would you like more detailed explanations for each property or have any other questions?

Further Questions:

1. What are the main differences between directed and undirected graphs?
2. How can a relation be both symmetric and antisymmetric?
3. What changes would make this graph reflexive?
4. Can a relation be transitive but not reflexive?
5. How can equivalence relations be visualized on a graph?

Tip:

When analyzing directed graphs for relations, start by checking for reflexivity (self-loops) since it simplifies the identification of partial orders and equivalence relations.