This task is about equivalence classes. One has to describe the equivalence relations for the following equivalence relations:
$ R1={(a,b)∈R×R∣∣a∣=∣b∣} \\ R2={(a,b)∈Z×Z∣∃z∈Z : a−b=z⋅p} \quad $
for a given constant $p \in \mathbb {N}$
Can someone explain me this more clearly. I don't really understand what equivalence relations are.
Equivalence class are in somehow a new definition of "equality".
For example, $R_1$ is an equivalence relation on $\mathbb R$, and tw element are "the same", if they have the same norme. So for example, $1R_1(-1)$, $3R_1(-3)$... or to to visualize a bite better, instead of writing $(-1)R_11$, just imagine that it means that $-1=1$ but this $=$ is not the same than the one you know.
For $R_2$, it's an equivalence relation on $\mathbb Z$. And in $(\mathbb Z,R_2)$ elements are "the same" if $p\mid a-b$. So for example $p=2p=...=-4p$ where the $=$ has to be understood as the equality in $(\mathbb Z,R_2)$ and not as the usual equality. That why we rather write $pR_22pR_2...R_2 -4p$.