Consider the relation R on the set of integers such that (a, b) ∈ R if and only if the difference of a and b is divisible by m. It can be shown that R is an equivalence relation. Find the equivalence class of the integer 8 when m = 6.
$m|a-b$
So far I have shown that this is a equivalence by:
Reflexive: $ m|a-a \rightarrow (a,a)\in R.$
Symmetric: $m|a-b \leftrightarrow m|-(a-b) = m|b-a$, so this is symmetric.
Transitive: Say $m|a-b \wedge m|b-c \rightarrow m|b-a$, this is transitive.
But I am clueless how to find the equivalence class of integer 8 when m = 6.
I appreciate the help!
So every element $x$ such that $6\mid x-8$ is in equivalence class of $8$. So $x-8 = 6k$ for some integer $k$ so $$x = 6k+8$$ and thus $$[8] = \{...-10,-4,2,8,14,...\}$$