   Chapter 8.5, Problem 7ES ### 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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# Define a relation R on Z, the set of all integers as follows: For every m , n ∈ Z , m R n     ⇔   every   prime   factor   of   m is a prime factor if  n . Is R a partial order relation? Prove or give a counterexample.

To determine

To check:

Whether R is a partial order relation or not.

Explanation

Given information:

For all m,n,mRn every prime factor of m is a prime factor of n.

Concept used:

A relation that is reflexive, antisymmetric, and transitive is called a partial order.

Calculation:

To verify anti-symmetricity:

The relation is not anti-symmetric.

Counter example:

Observe that, 2R4 because every prime factor of 2 is a prime factor of 4.

Also that 4R2 because every prime factor of 4 is a prime factor of 2.

But 24

That 2R4 and 4R2 but 24

So, the relation is not anti-symmetric.

To verify transitivity:

Suppose that r,s,t such that rRs and sRt

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