Question

To prove\({R^n}\) is reflexive for all positive integers \(n\).

Short answer

The relation \({R^n}\) is reflexive for all positive integers \(n\)is proved

Step-by-step solution

 Given

All positive integers are given here.

The Concept ofreflexive relation

A homogeneous binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself.In a graph of a reflexive relation, every node will have an arc back to itself. Note that irreflexive says more than just not reflexive.

Determine the relation

Using mathematical induction

The result is trivial for\(n = 1\)

Assume\({R^n}\)is reflexive then\((a,a) \in {R^n}\), for all\(a \in A\)and\((a,a) \in R\)

Thus,\((a,a) \in {R^n}^\circ R = {R^{n + 1}}\)for all\(a \in A\)

Therefore, by the principle of mathematical induction \({R^n}\) is reflexive for all positive integers \(n\).