Question

How can the directed graph of a relation \(R\) on a finite set \(A\) be used to determine whether a relation is irreflexive?

Short answer

The corresponded digraph must have no loop.

Step-by-step solution

Given data

The directed graph is given.

Concept of irreflexive relation

A relation\(R\)on a set\(A\)is irreflexive if\((a,a) \notin R\)for every\(a \in A\).

Determine whether a relation is irreflexive

If a relation \({\rm{R}}\) on \(A = \left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\}\) is irreflexive.

Then, \(\left( {{a_i},{a_i}} \right) \notin R\).

Thus, the corresponded digraph must have no loop.