Graphical Representation of Relations

Relations can often be made more intuitive using graphs. In the graph, the elements of the set are represented by vertices (points), and the ordered pairs in the relation are depicted by edges (arrows) in the appropriate direction.

Formal Description

Let R be a relation on a set A. In the graphical representation:

• We draw a vertex for every a ∈ A.
• We draw an edge from a to b for every (a,b) ∈ R.
• If a = b, a self-loop starts from the vertex to itself.

Examples of Graphical Representation

Let A = {1,2,3}, R = { (1,2), (2,3), (3,1) }.

The graph will have three vertices: 1, 2, 3, with directed edges 1 → 2, 2 → 3, 3 → 1, forming a directed cycle.

Properties in Graph Form

• Reflexive relations have self-loops on every vertex.
• Symmetric relations have edges in both directions for every edge (if a → b, then b → a).
• Transitive relations require that chains of edges imply direct edges.
• Partial orders can be represented as directed acyclic graphs (DAGs).

Summary

Graphical representation helps in understanding and illustrating the properties of relations. Vertices represent elements, edges represent connections. This is particularly useful in mathematics, computer science, and network modeling.

Practice Exercise