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Math Help - Equivalence relation

  1. #1
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    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.
    Last edited by Plato; February 7th 2009 at 12:17 PM. Reason: Fix TeX
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  2. #2
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    Are you by any chance using Herbert Endertonís 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.
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