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Q59E

Expert-verifiedFound in: Page 583

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

All positive integers are given here.

**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.**

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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