Question

Suppose that the relation R is irreflexive. Is the relation \({R^2}\) necessarily irreflexive?

Short answer

No, \({{\rm{R}}^2}\) is not irreflexive.

Step-by-step solution

Given data

The relation \(R\) is irreflexive.

Concept used of irreflexiverelation

A relation \({\rm{R}}\) on a set \({\rm{A}}\) is called reflexive if \((a,a) \notin R\) for every element \(a \in A\).

Prove for irreflexive relation

Given \(R\) is irreflexive.

\((a,a) \notin R\)

\({{\rm{R}}^2}\)is not necessarily irreflexive.

For example, if \((a,b) \in R\) and \((b,a) \in R\), then:

\(\begin{array}{l}(a,a) \in R^\circ R = {R^2}\\(b,b) \in R^\circ R = {R^2}\\{\rm{Since, }}{R^*} = {R_1} \cup {R_2} \cup \ldots \cup {R_n}\\Therefore,\;{R^2} \subseteq {R^*}\\(a,a) \in {{\rm{R}}^2}\\(b,b) \in {{\rm{R}}^2}\end{array}\)

By the definition of reflexive, \({{\rm{R}}^2}\) is not then reflexive.