Given that $R=\left\{(a,b)|3dividesa-b\right\}$ is an equivalence relation in steel in the set of integers $Z$. What is the number of partitions of $Z$?

Given that:

$R=\left\{(a,b):3dividesa-b\right\}$

Case I : Reflexive

Since,  $a-a=0$

And                                              

$3divides0,\therefore \frac{0}{2}=0$

$\Rightarrow 2dividesa-a$

$\therefore (a,0)\in R.$

$\therefore R\text{is}\text{reflexive}$

Case II : - Symmetric

$\text{If}\text{3 divides a - b,}$

$\text{then, 3 divides - (a - b) i}\text{.e}\text{. b - a}$

 $\text{Hence, If ( a , b) }\in \text{R, Then ( b , a) }\in R$

 $\therefore R\text{is}\text{symmetric}\text{.}$

 $\text{Case}\text{III}\text{. Transitive}\text{.}$

 $\text{If 3 divides ( a - b) and 3 divides ( b - c)}$

 $\text{So, 2 divides ( a - b) + ( b - c)}$

 $\text{also,}$

 $\text{2 divides ( a - c)}$

 $\therefore If(a,b)\in R\text{and}(b,c)\in R$

 $then,$

 $(a,c)\in R$

 $\text{Thus, R is an equivalence relation in z}\text{.}$

Hence, this is the answer.