Directed Graphs (Digraphs)
A directed graph consists of nodes or vertices connected by directed edges or arcs. Let R is relation from set A to set B defined as (a,b) ? R, then in directed graph-it is represented as edge(an arrow from a to b) between (a,b).
Properties
- A relation R is reflexive if there is a loop at every node of a directed graph.
- A relation R is irreflexive if there is no loop at any node of directed graphs.
- A relation R is symmetric if, for every edge between distinct nodes, an edge is always present in the opposite direction.
- A relation R is asymmetric if there are never two edges in opposite directions between distinct nodes.
- A relation R is transitive if there is an edge from a to b and b to c, then there is always an edge from a to c.
Example:
The directed graph of relation R = {(a,a),(a,b),(b,b),(b,c),(c,c),(c,b),(c,a)} is represented as:
Since, there is loop at every node, it is reflexive but it is neither symmetric nor antisymmetric as there is an edge from a to b but no opposite edge from b to a and also directed edge from b to c in both directions. R is not transitive as there is an edge from a to b and b to c but no edge from a to c.
Representation of Relation in Graphs and Matrices
Representation of Relations in Graphs and Matrices: Understanding how to represent relations in graphs and matrices is fundamental in engineering mathematics. These representations are not only crucial for theoretical understanding but also have significant practical applications in various fields of engineering, computer science, and data analysis. This article will explore different ways to represent relations using graphs and matrices, their properties, and applications in engineering.