The quotient set is the result of an abstraction process in our mind: It does not play any role which element of an equivalence class $[a]$ we choose to regain the same class. It can be any element $x\in V$, we only have to make sure that $x\sim a$ and we achieve immediately $[x]=[a]$. This motivates the following definition:

Definition: Complete System of Representatives

Let $V/_R$ be a quotient set generated by a given equivalence relation $R\subseteq V\times V.$ The element $a\in V$, of which there is an equivalence class $[a]\in V/_R$, is called a representative of $[a]$. A set of representatives $Q=\{a\in V : [a]\in V/_R\}$ is called a complete system of representatives of $V/_R.$

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