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Topic: A intuitive notion of set size.
Replies: 2   Last Post: Jan 13, 2013 1:02 PM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: A intuitive notion of set size.
Posted: Jan 13, 2013 7:13 AM
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On Jan 13, 4:20 am, Zuhair <zaljo...@gmail.com> wrote:
> For any two sets A,B:
> [1] A is bigger than B iff  |A| > |B| Or A,B are sets of naturals and
> there exist a non empty set C of naturals such that for every element
> n of C:
> |A(n)| > |B(n)| and |A(n*)| / |B(n*)| >= |A(n)| / |B(n)|.
> where X(n) = {y| y in X & y <' n};
> <' stands for natural strict smaller than relation;
> | | stands for cardinality defined after Cantor's.
> n* stands for the immediate successor of n in C with respect to
> natural succession.
> [2] A is smaller than B iff B is bigger than A.
> [3] A is equinumerous to B iff ~ A bigger than B & ~ A smaller than
> B.
> /


Example:

For natural k, let n_k = 3*2^{k-1} + 1;
so n_1 = 4, n_2 = 7, n_3 = 13, n_4 = 25, and so on.

Partition N into two seta A and B as follows:

A = {1} u [n_1, n_2) u [n_3, n_4) u [n_5, n_6) u ...
= {1, 4, 5, 6, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 49,
50, ...};

B = {2, 3} u [n_2, n_3) u [n_4, n_5) u [n_6, n_7) u ...
= {2, 3, 7, 8, 9, 10, 11, 12, 25, 26, 27, 28, 29, 31, 32, ...}.

In other words: the first natural, 1, is in A: the next 2 naturals are
in B; the next 3 are in A, the next 6 in B, the next 12 in A, the next
24 in B, and so on.

By your definition, A is bigger than B.
Hint: let C = {n_2, n_4, n_6, n_8, n_10, ...}.

By your definition, B is bigger than A.
Hint: let C = {n_1, n_3, n_5, n_7, n_9, ...}.

So A is both bigger and smaller than B. B is both bigger and smaller
than A. Since A is equinumerous to itself, the relations "bigger than"
and "smaller than" fail to be transitive.

This does not seem very intuitive to me.



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