   Chapter 8.2, Problem 12ES ### 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.E is the congruence modulo 4 relation on Z: For every m , n ∈ Z , m   E   n ⇔ 4 | ( m − n ) .

To determine

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

Explanation

Given information:

E is the congruence modulo 4 relation on Z:

For all m, n ∈ Z, m E n ∈ 4|(mn).

Calculation:

Let us consider the given relation E as

E={m,nZ| 4|(mn)}

Reflexive:

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

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

We note that E contains (a,a) because aa=0 and 4 divides 0 (as 40=0 with 0 an integer) and thus E is reflexive.

Hence reflexive

Symmetric:

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

Let us assume that (a,b)E. by the definition of E :

4|(ab)

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

(ab)=4c

Multiply each side by − 1:

(ab)=4c

Distributive property:

ba=4(c)

This last inequality then implies that 4|(ba) since − c is an integer (as c is an integer), which implies (b,a)E and thus E is symmetric

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