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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 3:33 PM

On 1/6/2013 2:22 PM, fom wrote:
> 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,

Yep, missed a C

ZFC

>
> =================================
> "|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
>
>