Chapter 8.2, Problem 18ES

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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 Q on R as follows: For all real numbers x and y , x Q y   ⇔ x − y is rational.

To determine

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

Explanation

Given information:

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

Calculation:

Let us consider the set below.

Q={x,yR|xy is rational}

Reflexive:

The relation Q is reflexive if (a,a)Q for every element aR.

We note that Q contains (x,x) because xx=0 and 0 is rational (since 0=01), thus Q is reflexive.

xRx

It is reflexive

Symmetric:

The relation Q on a set R Is symmetric if (b,a)Q whenever (a,b)Q.

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

ab is rational

By the definition of rational, there exists integers c and d such that:

ab=cd

Multiply each side by − 1:

(ab)=cd

Distributive property:

ba=cd

This last inequality then implies that b − a is irrational since cd is rational (as − c is an integer and d is an integer), which then implies (b,a)Q and thus Q is symmetric

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