Question

Suppose that the relation R is reflexive. Show that \({R^*}\) is reflexive.

Short answer

Thus, \({R^*}\) is reflexive.

Step-by-step solution

Given data

The relation \(R\) is reflexive.

Concept used of reflexive relation

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

Prove for reflexive relation

Given \(R\) is reflexive.

Since \({R^*} = R \cup {R^2} \cup {R^3} \cup \ldots \cup {R^n}\), so

\(R \subseteq {R^*}\)

Let \(a \in A\). Since \({\rm{R}}\) is reflexive:

\((a,a) \in R\)

Since \(R \subseteq {R^*}\)

\((a,a) \in {R^*}\)

Then \({{\rm{R}}^*}\) has to be reflexive (since \((a,a) \in {R^*}\) for every element \(a \in A\) )

Thus, \({R^*}\) is reflexive.