Question

((x, y) E A x A Let, A (0, 1,2, 3, 4, 5, 6) and R x - y is divisible by 3) (a) Prove that R is reflexive.

FOR N = 3

ANSWER QUESTION 11.

The parts above it are the previous exercise it mentions. For N = 3

Answer 1

11.

R: {(x,y)$\epsilon$Z*Z : x-y divisible by n=3}

a)

now let y=x

so

relation x-y =x-x=0 and Zero is divisible by 3

so

relationship is follows for (x,x) Hence its reflexive

b)

now

if x-y divisible by n=3

so

-(x-y) =y-x is also divisible by 3

Hence

x R y => yRx

Hence its Symmetric

c)

let

there are 3 elements x,y,and u

x-y is divisible by 3

y-u is divisible by 3

now so

(x-y)+(y-u) =x-u is also divisible by 3

Hence this relation is transitive

d)

there are two partitions of Z*Z that is

[x] => {(x,y)$\epsilon$Z*Z : x-y divisible by n=3}

[x] =>{(x,y)$\epsilon$Z*Z : x-y not divisible by n=3}