The set R:
$R = $ {(0,1), (0,0), (1,0), (1,1), (2,3), (2,2), (3,2), (3,3) (4,5), (5,4), (4,4), (6,6), (5,5) (6,7), (7,6), (7,7)}
I was just wondering if I've found the right equivalence classes for this relation?
My working:
The relation is an equivalence relation. Therefore, $[X]_R = ${ {0,1}, {2,3}, {4,5}, {6,7} }
Yes. $$X/R = \{\{0,1\},\{2,3\},\{4,5\},\{6,7\}\}$$
The first task is to confirm that $R$ is an equivalence relation over $\{0,1,2,3,4,5,6,7\}$ (that is $X$). After your correction, we can quickly verify that $R$ is indeed reflexive, symmetric, and transitive over $X$. After a quick resort, we can just eyeball it.
$${R}~{=}~{\{~{{(0,0), (0,1), (1,0), (1,1),}\\{ (2,2), (2,3), (3,2), (3,3),}\\{ (4,4), (4,5), (5,4), (5,5),}\\{ (6,6), (6,7), (7,6), (7,7)}}~\}}$$
The next task is to identify what subsets sets of the base set are related. That is, determine how $R$ partitions the set. We only need to step through all of the eight elements of $X$ and sort them into sets of related elements. We immediately see that $0$ is related only to itself and $1$, and so forth.
Thus this partition is indeed: $\{\{0,1\},\{2,3\},\{4,5\},\{6,7\}\}$.
This partition is the set of the equivalence classes of $R$ over $X$.