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Topic:
Intuitive criterion for set size.
Replies:
1
Last Post:
Jan 12, 2013 4:32 PM




Re: Intuitive criterion for set size.
Posted:
Jan 12, 2013 4:32 PM


On Jan 12, 2:08 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 set C of naturals such that for every element n of C:  > A(n) > B(n) and A(n+1)  B(n+1) > 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.
According to this definition, if A and B are *any* sets of naturals, then A is bigger than B (and B is bigger than A). Perhaps, by "a set C of naturals", you meant "a nonempty set C of naturals"?
By the way, your condition "A(n+1)  B(n+1) > A(n) B(n)" is equivalent to the simpler statement "n is an element of A but not an element of B". By saying that this condition holds for every element n of C, you are in effect saying that C is a subset of A and is disjoint from B, i.e., C is a subset of the set difference A\B.
> [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.



