Re: Cardinality and injection

On 17 Nov 2005 09:46:23 -0800, "zuhair" <zaljohar@xxxxxxxxx> wrote:

>
>William Hughes wrote:
>> zuhair wrote:
>> > William Hughes wrote:
>>
>>
>> <snip>
>>
>>
>>
>> I write
>>
>>
>> > > if A is a subset of B and A and B have the same cardinality,
>> > > then subcard A <> subcard B, unless A=B
>>
>>
>> You reply
>>
>> > No A do not necessarily equal B.
>> >
>> > see set A = 1,2,3,4,5,...............
>> > set B = 1,1/2,1/3,1/4,1/5,................
>> >
>> > These are also set which has subcard A = subcard B
>> >
>> > Also set A = 1,2,3,4,5,............
>> > and set B = -1,-2,-3,-4,-5,..............
>> >
>> > subcard A = subcard B.
>>
>>
>>
>> Do you see the problem. (Hint, is A a subset of B
>> in either example?)
>>
>>
>> Now go back and answer the questions in the post.
>>
>> -William Hughes
>
>I am not understanding anything of what you are saying or hinting at.
>
>All of what I wanted to say is that if their are two sets A and B
>
>having an intersectional set A.B that is infinite and at the same time
>
>a proper subset of at least one of them , then subcard A <> subcard B.
>
>And if sets A is a subset of set B and subcard A <> subcard B.
>
>then A is a proper subset of B.
>
>Subcardinality can define Properness of subsethood and not
>
>subsethood itself.
>
>Given two sets were their subcardinalities are "<>" doesn't mean
>
>that one should be a subset of the other.
>
>But if we know that one is a subset of the other then we know
>
>that one of them should be a proper subset of the other
>
>That's all
>
>Zuhair

The relation "not equals" can already do that ...

If A is a subset of B and A is not equal to B, then a is a proper
subset of B.
.

Relevant Pages

  • Re: Cardinality and injection
    ... > zuhair wrote: ... >> subcard A = subcard B. ... > fmust be a proper subset of A, and hence a proper subset of B. ... A is a proper subset of B if every member x in A is member x in B, ...
    (sci.math)
  • Re: Cardinality and injection
    ... > zuhair wrote: ... >> William Hughes wrote: ... >> that one of them should be a proper subset of the other ... subcard A = subcard B. ...
    (sci.math)
  • Re: Cardinality and injection
    ... >> zuhair wrote: ... A was a subset of B. You replied with two counterexamples, ... > a proper subset of at least one of them, then subcard A subcard B. ...
    (sci.math)
  • Re: Cardinality and injection
    ... >> zuhair wrote: ... >>> having an intersectional set A.B that is infinite and at the same time ... >>> that one of them should be a proper subset of the other ... > subcard A = subcard B. ...
    (sci.math)
  • Re: Cardinality and injection
    ... > card A <= card B. ... the definition of 'proper subset' makes no mention of cardinality. ... > Then subcard A = subcard B ...
    (sci.math)