Question:
Let $R$ be a relation on the set $Z$ of all integers defined as $(x, y) \in \mathrm{R} \Leftrightarrow x-y$ is divisible by 2 . Then, the equivalence class [1] is _________________.
Solution:
Given: $R$ is the equivalence relation on the set $Z$ of integers defined as $(x, y) \in R \Leftrightarrow x-y$ is divisible by 2 .
To find the equivalence class [1], we put y = 1 in the given relation and find all the possible values of x.
Thus,
$R=\{(x, 1): x-1$ is divisible by 2$\}$
$\Rightarrow x-1$ is divisible by 2
$\Rightarrow x=\pm 1, \pm 3, \pm 6, \pm 9, \ldots .$
Therefore, equivalence class $[0]=\{\pm 1, \pm 3, \pm 6, \pm 9, \ldots .\}$c
Hence, the equivalence class [1] is $\{\underline{\pm} 1, \pm 3, \pm 6, \pm 9, \ldots\} .$