Question

on Z, R={(x,y): x-y is divisible by 5} check if the given relation is reflexive, symmetric or transitive

Answer 1

$\large\underline{\sf{Solution-}}$

On the set of integers, Z, Relation R is defined as

$\red{\rm :\longmapsto\:R = \{(x,y) : \: x - y \: is \: divisible \: by \: 5 \}}$

1. Reflexive

$\rm :\longmapsto\:Let \: a \: \in \: Z$

We know,

$\rm :\longmapsto\:0 \: is \: divisible \: by \: 5$

$\rm :\longmapsto\:(a - a) \: is \: divisible \: by \: 5$

$\bf\implies \:(a,a) \: \in \: R$

$\bf\implies \:R \: is \: Reflexive$

2. Symmetric

$\rm :\longmapsto\:Let \: a, \: b \: \in \: Z$

such that

$\rm :\longmapsto\:(a,b) \: \in \: R$

$\rm :\longmapsto\:a - b \: is \: divisible \: by \: 5$

$\rm :\longmapsto\:b - a \: is \: divisible \: by \: 5$

$\bf\implies \:(b,a) \: \in \: R$

$\bf\implies \:R \: is \: symmetric$

3. Transitive

$\rm :\longmapsto\:Let \: a, \: b, \: c \: \in \: Z$

such that,

$\rm :\longmapsto\:(a,b) \: \in \: R$

$\rm :\longmapsto\:a - b \: is \: divisible \: by \: 5$

$\rm :\longmapsto\:a - b \: = \: 5x - - - (1)$

and

$\rm :\longmapsto\:(b,c) \: \in \: R$

$\rm :\longmapsto\:b - c \: is \: divisible \: by \: 5$

$\rm :\longmapsto\:b - c \: = \: 5y - - - (2)$

On adding equation (1) and equation (2), we get

$\rm :\longmapsto\:a - c \: = \: 5(x + y)$

$\rm :\longmapsto\:a - c \: is \: divisible \: by \: 5$

$\bf\implies \:(a,c) \: \in \: R$

$\bf\implies \:R \: is \: transitive$

Basic Concept :-

Let R be a relation defined on set A then

1. Relation R is Reflexive if (a, a) ∈ R for all a ∈ A.

2. Relation R is symmetric if (a, b) ∈ R then (b, a) ∈ R for all a, b ∈ A.

3. Relation R is transitive if (a, b) ∈ R, (b, c) ∈ R then (a, c) ∈ R for all a, b, c ∈ A