Representation of Relations using Graph

A relation can be represented visually using a directed graph (digraph). This graphical representation makes it easy to understand which elements are related and in what direction.

How to Represent a Relation as a Graph

The rules for converting a relation into a directed graph are −

• The number of vertices equals the number of elements in the set.
• For each ordered pair (x, y) in the relation R, draw a directed edge from vertex x to vertex y.
• If there is an ordered pair (x, x), draw a self-loop on vertex x.

Example

Suppose there is a relation R = { (1, 1), (1, 2), (3, 2) } on set S = { 1, 2, 3 }. This can be represented by the following directed graph ?

In the graph above −

• The self-loop on vertex 1 represents the ordered pair (1, 1).
• The directed edge from 1 to 2 represents the ordered pair (1, 2).
• The directed edge from 3 to 2 represents the ordered pair (3, 2).
• Vertex 3 has no incoming edges and no self-loop, meaning no element maps to 3 in this relation.

Identifying Relation Properties from a Graph

The graph representation also helps identify properties of the relation at a glance −

Conclusion

A relation on a set can be represented as a directed graph where vertices are the set elements and directed edges represent ordered pairs. Self-loops indicate elements related to themselves. This visual representation makes it easy to identify relation properties like reflexivity, symmetry, and transitivity.