# Math Help - Equivalence relation

1. ## Equivalence relation

Will someone help me with this problem. I'm struggling with it.
Define an equivalence relation R on the set P of positive integers by $mRn \Longleftrightarrow$ m and n have the same number of prime factors. Is there a function $f:P/R \rightarrow P/R \text{ such that } f([n]_R)=[3n]_R$ for each n?
I don't know how to show if this function exists or not. I just learned that a function is a relation such that if the ordered pair (x,y) and (x,y') are elements in R then, y=y'. I don't know how to apply that here. Please help.

2. Are you by any chance using Herbert Enderton’s text ELEMENTS OF SET THEORY?
If so, this problem appears in it with a nice theorem about function compatible with equivalence relations.
If not, here is the basic idea. Consider the function $f:\mathbb{Z} \mapsto \mathbb{Z}\;;\;f(n) = 3n$.
It should be clear to you that if $j\;\& \;k$ have the same number of prime factor then so do $3j\;\& \;3k$.
In other words, $jRk\; \Rightarrow \;f(j)Rf(k)$ this is enough to say $f$ is compatible with $R$.
Thus there exists $F:\left\{ {\left[ n \right]_R :n \in \mathbb{Z}} \right\} \mapsto \left\{ {\left[ n \right]_R :n \in \mathbb{Z}} \right\}\;;\;F\left( {\left[ n \right]_R } \right) = \left[ {f(n)} \right]_R$.

If you need details your library should a copy of that text.