   Chapter 8.2, Problem 19ES ### 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. Define a relation I on R as follows: For all real numbers x and y , x | y ⇔ x − y is irrational.

To determine

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

Explanation

Given information:

Define a relation I on R as follows: For all real numbers x and y, x I y ⇔ x - y is irrational.

Calculation:

Let us consider the set below.

I={x,yR|xy is irrational}

Reflexive:

The relation I is reflective if (a,a)I for every element aR.

We note that I does not contain (x,x) because xx=0 and 0 is rational (since 0=01), thus I is not reflexive.

It is not reflexive

Symmetric:

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

Let us assume that (a,b)I.

By definition of I :

ab is irrational

Let us assume that b − a is rational, by the definition of rational, there exists integers c and d such that:

ba=cd

Multiply each side by − 1:

(ba)

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