Define a relation $R$ on the set of all integers $Z$ by $xRy$ ($x$ related to $y$) if and only if $x-y=3k$ for some integer $k$.
I have already verified that this is in fact an equivalence relation. But now I need to find how many distinct equivalence classes there are.
I am confused on how to find how many distinct equivalence classes there are.
Best Answer
All equivalence classes are congruent classes of modulo 3. $Cl(0)=\{0, 3, -3, 6, -6,\ldots\}$ $Cl(1) = \{1, -2, 4, -5, 7, \ldots\}$ $Cl(2)=\{2, -1, 5, -4,\ldots\}$