We define a relation on set of Whole Numbers W0123 As follow

We define a relation on set of Whole Numbers

W={0,1,2,3…} As follows:

We divide the number by 5 and look at the remainder. All numbers that result into the same remainder are related.

E.g. 2/5 remainder =2, 7/5 remainder =2 and 12/5 remainder =2, therefore 2R7, 7R2, 7R12, 12R7, 12R2, and 2R12 etc.

[2] = {2,7,12,17….}

How many Equivalence classes are there.

Write the first four members of each equivalence class

Solution

Given whole number set {0,1,2,3........}

The numbers divides by 5

Any number divided by 5 we get 0,1,2,3,4 are only the remainders.

The number which is exactly divisble by 5 we get remainder 0.

The number which is not divisible by 5 we get the remainders(1,2,3or4).

Since the given set is a whole numbers set ..the divisble numbers of 5 are greater than five and 5.that mean the whole numbers from5,6,7....are divisible by 5.

There are five equivalence classes for 5.

The frist four members of each equivalence is

[0]={0,5,10,15...}

[1]={1,6,11,16......}

[2]={2,7,12,17......]

[3]={3,8,13,18.......}

[4]={4,9,14,19.......}.

For zero remainder

5/5=0,10/5=0,….

Therefore{0,5,10,15.....}

For 1

6/5=1,11/5=1...

For2

7/5=2,12/5=2,...

For 3

8/5=3,13/5=3.....

For 4

9/5=4,14/5=4,.....