For any equivalence relation $\sim$ on vector space $X$, is there always a map $f:X\rightarrow R$ such that $f(x)=f(y) \quad iff \quad x \sim y$? How to construct such maps?
I am just wondering how to prove DEFINITION 2 (I think it is not a definition at all) in this paper.
If you take $R'=\{[x]\colon x\in X\}$ to be the set of equivalence classes of $\sim$, then you can take $f\colon X\to R'$ to be the function $f(x) = [x]$ (the image is the equivalence class containing $x$).
So equivalence relations on $X$, partitions of $X$, and labeling functions from $X$ are all the same thing in different forms.
If you specifically mean for $R=\Bbb R$ to be the set of real numbers, then the answer is no. More precisely, the answer is yes precisely when the cardinality of the set $R'$ of equivalence classes is at most the cardinality of the reals (since then there exists an injective map from $R'$ to $\Bbb R$, and one can compose it with the $f$ defined above).
However, this cardinality inequality is not always satisfied. For example, if $X=\mathcal P(\Bbb R)$ is the set of all subsets of the real numbers and the equivalence relation in question is simply equality of subsets, then there is no such map $f$ because there is no injective function from $\mathcal P(\Bbb R)$ to $\Bbb R$.