Description of equivalence classes

Question

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.

There are two ways to describe equivalence classes. One, which is given here, is to state the conditions under which $a$ is equivalent to $b$. The other is to state each equivalence classes' contents. Can you see, for each of the two given problems, what are in the equivalence classes? — J.G., Nov 02 '21 at 10:23

score 1 · Answer 1 · answered Nov 02 '21 at 10:24

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 $pR_22pR_2...R_2 -4p$.

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