Let R be the equivalence relation on the set Z of integers given by
Question:

Let R be the equivalence relation on the set Z of integers given by R = {(a, b): 3 divides a-b}. Then the equivalence class [0] is equal to ____________________.

Solution:

Given: R is the equivalence relation on the set Z of integers given by R = {(a, b): 3 divides a − b}.

To find the equivalence class [0], we put b = 0 in the given relation and find all the possible values of a.

Thus,

$R=\{(a, 0): 3$ divides $a-0\}$

$\Rightarrow a-0$ is a multiple of 3

$\Rightarrow a$ is a multiple of 3

$\Rightarrow a=3 n$, where $n \in Z$

$\Rightarrow a=0, \pm 3, \pm 6, \pm 9, \ldots .$

Therefore, equivalence class $[0]=\{0, \pm 3, \pm 6, \pm 9, \ldots .\}$

Hence, the equivalence class $[0]$ is equal to $\{\underline{0}, \pm 3, \pm 6, \pm 9, \ldots\}$.