Concrete explanation for how an equivalence relation is related to equivalence class and the notation employed

Question

I'm fairly comfortable with the definition of what the three equivalence relations are. What I'm not comfortable and finding it above my head is how equivalence relation is closely related to equivalence class and the associated notation use to communicate the ideas.

Allow me to be be more specific.

Let $$\sim _n$$ denote an equivalence relation. For each $$\text{a$\epsilon $Z}$$, we let $$\text{[a}]_n$$ denote the equivalence class containing a. What does the text in italic implies in a concrete sense? Thanks in advance

Answer 1

Say we have some equivalence relation $\sim_n$ on a set $Z$. Then if two elements $a,b \in Z$ are related through the relation $\sim_n$, we write $a \sim_n b$. From this, we can talk about the equivalence class of an element $a \in Z$ with respect to the relation $\sim_n$ and we denote this $[a]_n$. An equivalence class is simply all the elements that are related to $a$ through $\sim_n$, or more formally $$[a]_n = \{b \in Z \mid a \sim_n b\}.$$ So the equivalence classes are constructed from the equivalence relation. Indeed, different equivalence relations on the same setwill give rise to different equivalence classes.

Answer 2

In general if $(E,\mathcal{R})$ is a set with an equivalence relation the collection of sets $E_x=\{y\in E, x\mathcal{R}y\}$ form a partition of $E$.

More specifically $E_x\neq \phi$ and for $x\neq y$ $E_x=E_y$ if $x\mathcal{R} y$ or $E_x\cap E_y=\phi$ otherwise