- #1
phoenixthoth
- 1,605
- 2
trying to define n/m when it exists in N.
suppose m and n are natural numbers and let [n,m] denote all functions from n (which is {0=Ø, 1, ..., n-1}) onto m. 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!
suppose m and n are natural numbers and let [n,m] denote all functions from n (which is {0=Ø, 1, ..., n-1}) onto m. 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!
Last edited: