   Chapter 8.2, Problem 46ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In 43-50, the following definitions are used: A relation on a set A is defined to be Irreflexive if, and only if, for every x ∈ A ,   x R x ; asymmetric if, and only if, for every x , y ∈ A if x R y then y R x ; intransitive if, and only if, for every x , y , z ∈ A , if x R y and y R z then x R z .For each of the relations in the referenced exercise, determine whether the relation I irreflexive, asymmetric, intransitive, or none of these.Exercise 4

To determine

To justify whether the relation is irreflexive, asymmetric, intransitive, or none of these.

Explanation

Given information:

A= {0,1,2,3}

R4={(1,2),(2,1),(1,3),(3,1)}

Calculation:

We have:

A= {0,1,2,3}

R4={(1,2),(2,1),(1,3),(3,1)}

Irreflexive:

The relation R4 is irreflective if (a,a)R4 for every element aA.

Since A={0,1,2,3},R4 is irreflexive if it does not contain (0,0),(1,1),(2,2),(3,3).

We note that R4 does not contain (0,0),(1,1),(2,2) nor (3,3) and thus R4 is irreflexive.

Asymmetric:

The relation R4 on a set A is asymmetric if (b,a)R4 whenever (a,b)R4.

We note that (1,2)R4 while (2,1)R4 and thus R4 is not asymmetric.

Intransitive:

The relation R4 on a set A is intransitive if (a,c)R4 whenever (a,b)R4 and (b,c)R4

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