Given this relation: $R=\{(a,b) \in \mathbb{Z}\times\mathbb{Z} \quad aRb \iff \exists h\in \mathbb{Z}: \quad b+3a=4h\}$ proves:
1) $R$ is an equivalence relation. Proved.
2) show the partition set $\mathcal{Z}_R$ inducted by $R$.
For the second question, I have already a solution (not from me) but I'm not sure that is correct: $$\mathcal{Z}_R = \mathbb{Z}^2/R =\{[0]_R,[1]_R\}$$ where:
$$[0]_R=\{b \in \mathcal{Z}; \quad 0Rb\}= \{b \in \mathcal{Z}; \quad \exists h \in \mathcal{Z}: b=4h \}$$
and
$$[1]_R=\{b \in \mathcal{Z}; \quad 1Rb\}= \{b \in \mathcal{Z}; \quad \exists h \in \mathcal{Z}: b+3=4h \}$$
Essentially I don't understand why only these 2 classes are in $\mathcal{Z}_R$. Why is this correct(if it is)? If not can someone explain me how can find the equivalence classes of the partition set. thanks in advance