   Chapter 8.2, Problem 17ES ### 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 9-33, determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answer. Recall that a prime number is an integer that is greater than 1 and has no positive integer divisor other than 1 and itself, (In particular, I is not prime.) A relation P is defined on Z as follows: For every m , n ∈ Z ,   m P n ⇔ ∃ a prime number p such that p | m and p | n .

To determine

To justify whether the given relation is reflexive, symmetric, transitive, or none of these.

Explanation

Given information:

A prime number is an integer that is greater than 1 and has no positive integer divisors other than 1 and itself. (In particular, 1 is not prime) A relation P is defined on Z as follows: For all m, n ∈ Z ,m P n ⇔ ∃ a prime number p such that p | m and p | n.

Calculation:

Reflexive:

P is not reflexive since 1 P 1 (notice that there is no prime number divides 1).

It is not reflexive

Symmetric:

P is symmetric since mPn implies nPm (since that if p divides m and n, then p divides n and m )

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