I am really upset as I am not able to understand the difference between Equivalence Class $[x]=\{y \in G \mid yEx\}$ and the terms like $[0], [1], [2]$ etc. for $\mathbb{Z}_3$.
Please help me to understand the real meaning of these two.
2026-04-02 17:40:00.1775151600
Difference between equivalences classes and congruence classes
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Each element $[a]$ in $\mathbb{Z}_n$ is an equivalence class. To be precise, $[a]$ is the set of all integers $x$ which are congruent to $a \mod n$. So, $$ [a] = \{x \in \mathbb{Z} \mid x \equiv a \mod n\}. $$ The right-hand side is the literal definition of the equivalence class of $a$ when the relation is specified to congruence mod $n$.