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Topic: A size criterion: a question
Replies: 15   Last Post: Jan 9, 2013 1:22 PM

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 fom Posts: 1,968 Registered: 12/4/12
Re: A size criterion: a question
Posted: Jan 6, 2013 3:22 PM

On 1/6/2013 1:35 PM, Zuhair wrote:
> If we characterize cardinality in the following manner, How much that
> would differ from the known cardinality:
>
> |x| < |y| iff there exist an injection from x to y and there do not
> exist a surjection from a subset of x to y.
>
> |x| > |y| iff there exist a surjection from a subset of x to y and
> there do not exist an injection from x to y.
>
> |x| = |y| iff there exist an injection from x to y and there exist a
> surjection from a subset of x to y.

it appears to be equivalent

from Jech on ZF,

=================================
"|x| = |y| if there exists a one-to-one mapping of X onto Y"

injection + surjection
===================================
"|x| <= |y| if there is a one-to-one mapping of X into Y"

injection, surjection undetermined
====================================
"|x| < |y| <-> ((|x| <= |y|) /\ -(|x| = |y|))"

injection, no surjection