Let A1 2 3 4 5 6 7 9 and define R by xRy iff xy4k for some k

Let A={1, 2, 3, 4, 5, 6, 7, 9} and define R by xR_y iff x-y=4k for some k belongs to Z Is R an equivalence relation on A? Justify your answer. If R is an equivalence relation find A mod R, A/R

Solution

1. xRx as x-x=0 =4*0

2. Let, xRy so x-y=4k

HEnce, y-x=4*(-k)

So, yRx

3. Let, xRy and yRz

SO, x-y=4k, y-z=4m

x-y+y-z=x-z=4(k+m)

Hence, xRz

SO, R is reflexive,symmetric and transitive

So, R is equivalence relation

In other words xRy if they given same remainder on division by 4

So,

[0]={4}

[1]={1,5,9}

[2]={2,6}

[3]={3,7}

A mod R={[1],[2],[3],[4]}