   Chapter 8.5, Problem 21ES ### 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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# Consider the “divides” relation defined on the set A = { 1 , 2 , 2 , 2 , .....2 n } , where n is a nonnegative integer. Prove that this relation is a total order relation on A. Draw the Hasse diagram for this relation for n = 4.

To determine

(a)

To Prove:

The “divides | ” relation is a total order relation on given set:

A={1,2,22,23,......,2n}.

Explanation

Given information:

A={1,2,22,23,......,2n}

Concept used:

If given set is reflexive, antisymmetric and transitive then the given set is called of partial order.

Proof:

| Is reflexive: suppose aA. Then a=1a and 1=A so a|a.

| Is antisymmetric: For a,bA suppose a|b & b|a. Then there exist two integers k1,k2A such that b=k1a,a=k2b.

b=k1a=k1k2b

Since b0,k1k2=1 and hence k1=k2=±1.

Since a,bA,a,b are two positive integers k1k2>0.

Then k1=k2=1A

Therefore, a=b

| Is transitive: For a,b,cA suppose that a|b,b|c

To determine

(b)

To draw:

The Hasse diagram for this relation for n=4.

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