Question

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

Short answer

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

Step-by-step solution

Given

\(R\) is reflexive

Concept of Reflexive

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

Check the Reflexive

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

Given that\(R\)is reflexive.

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

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

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

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

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

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

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

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