Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 \(n\).

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Discrete Mathematics and its Applications

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Short Answer

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

Step by Step Solution

Step 1: Given

Step 2: The Concept ofreflexive relation

Step 3: Determine the relation

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Discrete Mathematics and its Applications

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Short Answer

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

Step by Step Solution

Step 1: Given

Step 2: The Concept ofreflexive relation

Step 3: Determine the relation

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