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Re: Date for Last Common Ancestor?
Stephen Barnard (steve@megafauna.com)
Wed, 21 Aug 1996 07:43:11 -0800
CHESSONP wrote:
>
> Help me flush out the model.
>
> Let H be the set of all human beings alive at time t_0. Let M_0 be
> the set of mothers (living or dead) of the members of H. Represent the
> functional relationship between H and M_0 by the symbol f, defined by
>
> m=f(h) means that m "is the mother of" h.
>
> The function so defined, f:H -->M_0, is onto, meaning that every element
> in M_0 has a "preimage" in the set H. Since H is finite, this implies
> logically that M_0 is finite, and, indeed, that the size of M_0 is no
> greater than the size of H.
>
> With this functional relationship we can define a sequence of sets
> {M_k} for all positive numbers, k, by defining:
>
> "m is an element of M_k, if, and only if, there exists an
> element d
> of the set M_k-1, such that m=f(d)."
>
> About the sets M_k, we know only that the size of M_k is no greater
> than the size of M_k-1. We will need further assumptions to get the
> sequences of sizes of {M_k} to converge to one.
>
> I apologize for the clumsiness of the notation, but without a decent
> symbol set I think this is the best I can do.
Your definition is fine, as far as it goes. Like you say, you need further
assumptions to get the size to converge to one.
The assumption that I made, which seems perfectly reasonable to me, is that the
probability that every mother in M_k-1 produces *exactly* one mother in M_k is
less than one.
Steve Barnard
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