   Chapter 8.3, Problem 20ES ### 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 19—31, (1) prove that the relation is an equivalence relation, and (2) describe the distinct equivalence classes of each relation. 20. E is the relation defined on Z as follows: For every m ,   n ∈ Z ,   m   E   n ⇔ 4 | ( m − n ) .

To determine

To prove:

(1) The given relation is an equivalence relation,

(2) Describe the distinct equivalence classes of each relation.

Explanation

Given information:

E is the relation defined onZ as follows: For all m, n ∈Z ,

m E n ⇔ 4 | ( m - n).

Proof:

(1). A relation will be equivalence relation if it is reflexive, symmetric and transitive.

To proof: E is an equivalent relation

A=ZE={(m,n)A×A|4|( mn)}

Reflexive:

Let xA=Z

xx=0=04

By the definition of divides, 4 divides x − x and thus (x,x)E.

Since (x,x)E for all xA and thus E is indeed reflexive.

Symmetric:

Let us assume that (x,y)E. by the definition of E :

4|(xy)

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

xy=4k

Multiply each side by − 1:

(xy)=(4k)

Use the distributive property:

yx=4(k)

By the definition of divides, 4 divides y − x as − k is an integer (because k is an integer).

(y,x)E

Since (x,y)E implies (y,x)E,E is symmetric.

Transitive:

Let us assume that (x,y)E and (y,z)E

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