   Chapter 1.7, Problem 4E

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# Let R be the relation “congruence modulo 5” defined on Z as follows: x is congruent to y modulo 5 if and only if x − y is a multiple of 5 , and we write x ≡ y ( mod 5 ) .a. Prove that “congruence modulo 5 ” is an equivalence relation.b. List five members of each of the equivalence classes [ 0 ] ,   [ 1 ] ,   [ 2 ] ,   [ 8 ] , and [ − 4 ] .

a)

To determine

To prove:

The relation “congruence modulo 5” defined on Z as x is congruent to y modulo 5 if and only if xy is a multiple of 5, written as xy(mod5) is an equivalence relation.

Explanation

Formula Used:

The relation “congruence modulo n” is defined on the set Z of all integers as follows:

x is congruent to y modulo n if and only if xy is a multiple of n. That is,

xy=nk for some kZ.

Proof:

A relation R on a nonempty set A is an equivalence relation if the following conditions are satisfied for arbitrary x, y, and z in A:

1. xRx for all xA. (Reflexive Property)

2. If xRy, then yRx. (Symmetric Property)

3. If xRy and yRz, then xRz. (Transitive Property)

Consider the relation xRy such that xy(mod5) if and only if xy is a multiple of 5 defined on Z.

1. xx(mod5), since xx=0=(5)(0).

2. xy(mod5)xy=5kforsomekZ

b)

To determine

The five members of each of the following equivalence classes ,,,, and .

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