   Chapter 8.2, Problem 14ES ### 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.O is the relation defined on Z as follows: For every m , n ∈ Z ,   m   O   n ⇔ m − n is odd.

To determine

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

Explanation

Given information:

O is the relation defined on Z as follows:

For all m,nZ, m O nmn is odd.

Calculation:

Let us consider the given relation O as

O={m,nZ|(mn) is odd}

Reflexive:

The relation O is reflexive if (a,a)O for every element aA.

O is reflexive if it contains (x,x) for all xZ.

We note that O does not contain (x,x) because xx=0 and 0 is not odd, thus O is not reflexive.

Thus, not reflexive

Symmetric:

The relation O on a set A is symmetric if (b,a)O whenever (a,b)O.

Let us assume that (a,b)O. By definition of O

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