   Chapter 8.3, Problem 26ES ### 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 dasses of each relation.D is the relation defined on Z as follows: For every m ,   n   ∈ Z m D n       ⇔   3 | ( m 2 − n 2 ) .

To determine

To prove:

(1) The given relation is an equivalence relation,

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

Explanation

Given information:

D is the relation defined on Z as follows: For all m, n ∈ Z ,

m D n ⇔ 3 | ( m2− n2).

Proof:

(1). To prove: R is an equivalent relation

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

Reflexive:

D is a relation on for m,n.

mDn3/m2n2

For any m,

m2m2=0=3/m2n2mDmD is reflexive.

Symmetric:

For m,n such that mDn.mDn3/m2n2m2n2=3k for some kn2m2=3(k)                          k3/n2m2nDm

D is symmetric.

For m,n,p such that mDn, nDp. Therefore:3/m2n2   &    3/n2p2m2n2=3k    &    n2p2=3l         for some k,lm2n2+n2p2=3k+3lm2p2=3(k+l)        &     k+lmDpD is transitive

So, D is reflexive, symmetric and transitive

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