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Expert-verified Found in: Page 583 ### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095 # To prove$${R^n}$$ is reflexive for all positive integers $$n$$.

The relation $${R^n}$$ is reflexive for all positive integers $$n$$is proved

See the step by step solution

## Step 1:  Given

All positive integers are given here.

## Step 2: 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.

## Step 3: 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 $$n$$. ### Want to see more solutions like these? 