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Re: [PrimeNumbers] Re: Is 2 a prime

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  • Peter Kosinar
    ... It depends on the area people work in. For a combinatorian, the equality 0^0 = 1 can work perfectly well; as the left-hand-side denotes the number of
    Message 1 of 15 , Aug 7, 2006
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      > Sorry, if I did not post right. I was not about to cause any
      > inconvenience.
      >
      > I did not say Jack's definition is not possible. I think is not the
      > definition mostly used, as usally division by zero is not defined. As
      > well as 0^0, if I remember right. Jack's definition leads to an
      > agreeable result, I admit.

      It depends on the area people work in. For a combinatorian, the equality
      0^0 = 1 can work perfectly well; as the left-hand-side denotes the number
      of functions from empty set to empty set [*]. Moreover, things like
      binomial theorem also work nice with this extension; it allows you to
      evaluate the sum [k=0,n,(-1)^k*(n choose k)] as being equal to (1-1)^n, or
      simply 0^n (ok, I admit, this is just a contrived academic example).

      Likewise, if you work in the area of foundations of mathematics, defining
      divisibility using the operation of division is a bit more complicated
      than using the straight existential-quantifer with multiplication (just
      like Jack did); for the division is only a derived operation in e.g. Peano
      Arithmetics and one needs to prove its well-definedness (and possibly some
      other properties) first.

      On the other hand, an analyst would probably bop you over the head
      if he saw 0^0 :-)

      Peter

      [*] This works even in the much more general framework of set-theory --
      If A and B are sets with cardinalities |A| resp. |B|, |A|^|B| is defined
      to be the cardinality of the set A^B which is the set of all functions
      from B to A. If the sets A and B are finite, the cardinal exponentation
      agrees with the usual exponentation of natural numbers.

      --
      [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
    • Dr. Michael Paridon
      So I thank everybody very much for explanation. Mathematics has more hidden beauties I will ever learn, and as I am not in a professional way dealing with
      Message 2 of 15 , Aug 7, 2006
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        So I thank everybody very much for explanation. Mathematics has more hidden
        beauties I will ever learn, and as I am not in a professional way dealing
        with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

        Best regrads

        Michael


        ----- Original Message -----
        From: "Peter Kosinar" <goober@...>
        To: "Dr. Michael Paridon" <dr.m.paridon@...>
        Cc: <primenumbers@yahoogroups.com>
        Sent: Monday, August 07, 2006 4:19 PM
        Subject: Re: [PrimeNumbers] Re: Is 2 a prime


        > > Sorry, if I did not post right. I was not about to cause any
        > > inconvenience.
        > >
        > > I did not say Jack's definition is not possible. I think is not the
        > > definition mostly used, as usally division by zero is not defined. As
        > > well as 0^0, if I remember right. Jack's definition leads to an
        > > agreeable result, I admit.
        >
        > It depends on the area people work in. For a combinatorian, the equality
        > 0^0 = 1 can work perfectly well; as the left-hand-side denotes the number
        > of functions from empty set to empty set [*]. Moreover, things like
        > binomial theorem also work nice with this extension; it allows you to
        > evaluate the sum [k=0,n,(-1)^k*(n choose k)] as being equal to (1-1)^n, or
        > simply 0^n (ok, I admit, this is just a contrived academic example).
        >
        > Likewise, if you work in the area of foundations of mathematics, defining
        > divisibility using the operation of division is a bit more complicated
        > than using the straight existential-quantifer with multiplication (just
        > like Jack did); for the division is only a derived operation in e.g. Peano
        > Arithmetics and one needs to prove its well-definedness (and possibly some
        > other properties) first.
        >
        > On the other hand, an analyst would probably bop you over the head
        > if he saw 0^0 :-)
        >
        > Peter
        >
        > [*] This works even in the much more general framework of set-theory --
        > If A and B are sets with cardinalities |A| resp. |B|, |A|^|B| is defined
        > to be the cardinality of the set A^B which is the set of all functions
        > from B to A. If the sets A and B are finite, the cardinal exponentation
        > agrees with the usual exponentation of natural numbers.
        >
        > --
        > [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
        >
        >
        > __________ NOD32 1.1694 (20060805) Information __________
        >
        > Diese E-Mail wurde vom NOD32 antivirus system gepr├╝ft
        > http://www.nod32.com
        >
        >
      • Alan McFarlane
        ... Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction. ... [snip] -- Alan
        Message 3 of 15 , Aug 7, 2006
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          Dr. Michael Paridon wrote:
          > So I thank everybody very much for explanation. Mathematics has more hidden
          > beauties I will ever learn, and as I am not in a professional way dealing
          > with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

          Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction.

          > Best regrads
          >
          > Michael
          >
          >
          > ----- Original Message -----
          [snip]


          --
          Alan
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