Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

Let $Z$ be the set of all integers and let $R$ be a relation on $Z$ defined by a $R b \Leftrightarrow(a-b)$ is divisible by $3 .$ Then, $R$ is

A. reflexive and symmetric but not transitive

B. reflexive and transitive but not symmetric

C. symmetric and transitive but not reflexive

D. an equivalence relation

Solution:

According to the question,

Given set $Z=\{1,2,3,4 \ldots . .\}$

And $R=\{(a, b): a, b \in Z$ and $(a-b)$ is divisible by 3$\}$

Formula

For a relation $R$ in set $A$

Reflexive

The relation is reflexive if $(a, a) \in R$ for every $a \in A$

Symmetric

The relation is Symmetric if $(a, b) \in R$, then $(b, a) \in R$

Transitive

Relation is Transitive if $(a, b) \in R \&(b, c) \in R$, then $(a, c) \in R$

Equivalence

If the relation is reflexive, symmetric and transitive, it is an equivalence relation.

Check for reflexive

Consider, $(a, a)$

$(\mathrm{a}-\mathrm{a})=0$ which is divisible by 3

$(a, a) \in R$ where $a \in Z$

Therefore , R is reflexive ……. (1)

Check for symmetric

$\therefore(\mathrm{a}-\mathrm{b})$ which is divisible by 3

- $(\mathrm{a}-\mathrm{b})$ which is divisible by 3

(since if 6 is divisible by 3 then $-6$ will also be divisible by 3 )

$\therefore(b-a)$ which is divisible by $3 \Rightarrow(b, a) \in R$

For any $(a, b) \in R ;(b, a) \in R$

Therefore , R is symmetric ……. (2)

Check for transitive

Consider , (a,b) ∈ R and (b,c) ∈ R

$\therefore(a-b)$ which is divisible by 3

and $(b-c)$ which is divisible by 3

$[(a-b)+(b-c)]$ is divisible by 3$]$ (if 6 is divisible by 3 and 9 is divisible by 3 then $6+9$ will also be divisible by 3)

$\therefore(\mathrm{a}-\mathrm{c})$ which is divisible by $3 \Rightarrow(\mathrm{a}, \mathrm{c}) \in \mathrm{R}$

Therefore $(a, b) \in R$ and $(b, c) \in R$ then $(a, c) \in R$

Therefore, $\mathrm{R}$ is transitive ....... (3)

Now, according to the equations (1), (2), (3)

Correct option will be (D)