   Chapter 8.2, Problem 13ES ### 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.F is the congruence modulo 5 relation on Z: For every m , n ∈ Z , m   F   n ⇔ 5 | ( m − n ) .

To determine

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

Explanation

Given information:

F is the congruence modulo 5 relation on Z:

For all m, n ∈ Z, m F n ⇔ 5|(mn).

Calculation:

Let us consider the given relation F as

F={m,nZ|5(mn)}

Reflexive:

The relation F is reflective if (a,a)F for every element aA.

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

We note that F contains (x,x) because xx=0 and 5 divides 0 (as 50=0 with 0 an integer) and thus F is reflexive.

Hence reflexive

Symmetric:

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

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

5|(ab)

By the definition of divides, there exists an integer c such that:

(ab)=5c

Multiply each side by − 1:

(ab)=5c

Distributive property:

ba=5(c)

This last inequality then implies that 5|(ba since) c is an integer (as c is an integer), which implies (b,a)F and thus F is symmetric

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