Name: Commutative & Associative Property of Binary Operation
Uploaded: 2019-07-19
Duration: 5 min 47 s
Description: To know more about the properties of Binary operations, watch the video and explore the different problems presented. Commit the concept to your memory with...

Question

Prove that the relation

R

in the set

A

=

(5, 6, 7, 8, 9}

, given by

R = {(a, b) : |a - b|

is divisible by

2}

, is an equivalence relation. Find all elements related to element

6

.

Accepted Answer

Given set $A = {5, 6, 7, 8, 9}$

and $R = {(a, b) : | a - b |$ is divisible by $6$ }

Reflexive : $\forall a \in A$ , since $| a - a | = 0 = 2 \cdot 0$

$\Rightarrow | a - a |$ is divisible by $2 \Rightarrow (a, a) \in R$

$∴$ $R$ is reflexive.

Symmetric: $\forall a, b \in A, (a, b) \in R$

$\Rightarrow | a - b |$ is divisible by $2$

$\Rightarrow a - b$ is divisible by $2$

$\Rightarrow a - b = 2 m$ , where $m \in Z$

$\Rightarrow b - a = - 2 m = 2 (- m)$ , where $- m \in Z$

$∴ b - a$ is divisible by $2$

$\Rightarrow | b - a |$ is divisible by $2$

$\Rightarrow (b, a) \in R \Rightarrow R$ is symmetric

Transitive $: \forall a, b, c \in A, (a, b) (b, c) \in R$

Now $(a, b) \in R \Rightarrow | a - b |$ is divisible by $2$

$\Rightarrow a - b$ is divisible by $2 \Rightarrow a - b = 2 m_{1} \dots (1)$ and $(b, c) \in R \Rightarrow | b - c |$ is divisible by $2$

$\Rightarrow b - c$ is divisible by $2$

$\Rightarrow b - c = 2 m_{2}$ ..... $(2)$

Where $m_{1}, m_{2} \in Z$

On adding $(1)$ and $(2)$ ; we have $a - b + b - c = 2 (m_{1} + m_{2})$

$\Rightarrow a - c = 2 m$ , where $m \in Z$

$\Rightarrow a - c$ is divisible by $2$

$\Rightarrow | a - c |$ is divisible by $2$ .

$\Rightarrow (a, c) \in R \Rightarrow R$ is transitive.

Thus $R$ is reflexive, symmetric and transitive.

Hence $R$ be an equivalence relation on $A$ .

To find elements related to $6$ :

Consider any $a \in A, (a, 6) \in R$

$\Rightarrow | a - 6 |$ is divisible by $2$

$\Rightarrow a - 6$ is divisible by $2$

$\Rightarrow a - 6 = 2 m, m \in I \Rightarrow a = 2 m + 6, m \in I$

Putting $m = 0, 1, 2, 3, \dots, - 1, - 2, - 3, \dots$

$a = 6, 8, 10, ∣ 2, \dots, 4, 2, 0, \dots$

But $a \in A = {5, 6, 7, 8, 9}$

$\Rightarrow a = 6, 8$

$∴$ The set of elements related to $6$ are $\{6, 8\}$ .

Prove that the relation $R$ in the set $A$ = $(5, 6, 7, 8, 9}$ , given by $R = {(a, b) : |a - b|$ is divisible by $2}$ , is an equivalence relation. Find all elements related to element $6$ .

Important Questions on Relations and Functions

Learn and Prepare for any exam you want !

Prove that the relation R in the set A= (5,6,7,8,9}, given by R={(a,b):a-b is divisible by 2}, is an equivalence relation. Find all elements related to element 6.

Important Questions on Relations and Functions

Learn and Prepare for any exam you want !

Prove that the relation $R$ in the set $A$ = $(5, 6, 7, 8, 9}$ , given by $R = {(a, b) : |a - b|$ is divisible by $2}$ , is an equivalence relation. Find all elements related to element $6$ .