   Chapter 8.3, 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 each of 7-14, the relation R is an equivalence relation on set A. Find the distinct equivalence classes of R.A is the set of all string of length 4 in a’s and b’s R is defined on A as follows: For all strings s and t in A, s R t ⇔   s   has   the ame first two characters as  t .

To determine

To find the distinct equivalence classes of R.

Explanation

Given information:

The relation R is an equivalence relation on the set A.

A is the set of all strings of length 4 in a’s and b’s. R is defined on A as follows: For all strings s and t in A ,

s R t       s has the same first two characters as t.

Calculation:

A= Set of all strings of length 4 in a's and b'sR={(s,t)A×A|s and t have the same first two characters}

Let us first determine all strings of length 4 in a’s and b’s :

A={aaaa,aaab,aaba,aabb,        abaa,abab,abba,abbb,        baaa,bbab,bbba,bbbb,        bbaa,bbab,bbba,bbbb}

Let us next group the ordered pairs in R that have at least one common element (with at least one of the other elements in the group).

(s,t)R

When s and t have the same first two characters.

First group: (aaaa,aaaa),(aaaa,aaab),(aaaa,aaba),(aaaa,aabb),                    (aaab,aaaa),(aaab,aaab),(aaab,aaba),(aaab,aabb),                    (aaba,aaaa),(aaba,aaab),(aaba,aaba),(aaba,aabb),                    (aabb,aaaa),(aabb,aaab),(aabb,aaba),(aabb,aabb)

Second group: (abaa,abaa),(abaa,abab),(abaa,abba),(abaa,abbb),                        (abab,abaa),(abab,abab),(abab,abba),(abab,abbb),                        (abba,abaa),(abba,abab),(abba,abba),(abba,abbb),                        (abbb,abaa),(abbb,abab),(abbb,abba),

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