I have done (a), pretty straight forward. I understand an equivalence class as all the elements in the domain that map to the same result in the co-domain. For example in (mod 3), [|0|] would be the same as [|3|] ect.
However I am a little confused in the notaiton here. The relation R clearly defines two coordinate pairs? I am rather confused on how to proceed with (b).
Any help/suggestions?
Thanks!
Substituting into our definitions, we have: \begin{align*} [(1, 0)] &= \{(m, n) \in \mathbb N_0 \times \mathbb N_0\mid ((m, n), (1, 0)) \in R\} \\ &= \{(m, n) \in \mathbb N_0 \times \mathbb N_0 \mid m + 0 = n + 1\} \\ &= \{(1, 0), (2, 1), (3, 2), (4, 3), \ldots\} \end{align*} Interpreting this graphically, $[(1, 0)]$ is the set of all lattice points in the first quadrant of the $xy$-plane that are on the line $y = x - 1$. The equivalence classes partition these lattice points by grouping them together along parallel lines of slope $1$.