trying to define n/m when it exists in N.(adsbygoogle = window.adsbygoogle || []).push({});

suppose m and n are natural numbers and let [n,m] denote all functions from n (which is {0=Ø, 1, ..., n-1})ontom. if [n,m] is empty, stop and say that m does not divide n.

consider the subset of functions f in [n,m] such that ~ defines an equivalence relation on n where x~y iff f(x)=f(y) and that each equivalence class has the same size q. call this set [[n,m]]. i'm guessing that if [[n,m]] is nonempty then it will be the same q for all functions in [n,m] such that ~ defines an equivalence relation partitioning n into equal sized parts (m parts each having q elements).

i suppose this is equivalent to saying

[[n,m]]={f in [n,m] : for all z in m, card(f^{-1}({z})) is constant}.

definition: if [[n,m]] is nonempty then let n/m=q. if [[n,m]] is empty say that m does not divide n.

question: n/m=q in this sense if and only if n/m=q in the usual sense? (i'll also have to decide if q is well defined.)

this is a definition of division not obviously equivalent to "the inverse of multiplication." one can now define multiplication to be the inverse of division!

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# Trying to define n/m when it exists in N.suppose m and n are

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