Date: Jan 6, 2013 3:22 PM
Author: fom
Subject: Re: A size criterion: a question

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