   Chapter 8.1, Problem 3ES ### 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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# The congruence modulo 3 relation, T, is defined from Z to Z as follows: For all integers m and n, ???????????3|???-?? Is 10 T 1? Is 2,2????? 8,1 ? ??? b. List five integers n such that n T0. List five integers n such that n T 1. List five integers n such that n T 2. Make and prove a conjecture about which integers are related by T to 0, which integers are related by T to 1, and which integers are related by T to 2.

To determine

(a)

Whether the given relations are true or not.

Explanation

Given information:

The congruence modulo 3 relation, T, is defined from Zto Z as follows: For all integers m and n, m T n ⇔ 3 | ( m - n).

The relations are 10 T 1 , 1 T 10 , (2,2)T, (8,1)T.

Calculation:

Let us consider the first relation below.

10 T 1

Here m=10 and n=1. Then mn=101=9 and

3|9 so 10 T 1 is true.

Let us consider the secondrelation below.

1 T 10

Here m = 1 and n = 10. (mn)=(110)=9 and

3|9 so 1 T 10 is true

To determine

(b)

List five integers n such that n T 0.

To determine

(c)

List five integers n such that n T 1.

To determine

(d)

List five integers n such that n T 2.

To determine

(e)

Make and prove a conjecture about which integers arerelated by T to 0, which integers are related by T to 1,and which integers are related by T to 2.

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