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 Empty sets

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nick
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PostSubject: Empty sets   Thu Sep 17, 2009 12:01 am

On one hand, there is a natural question that comes for a math student, and this is "How many empty sets are there ?" On the other hand, it is natural to work in maths with one single empty set.

Before you pay some attention to this subject, I advise you to choose other more constructive activity and let the fun logicians to spend their time here. I personally agree that one single empty set is enough to do good math, but it remains some inquiring about "what if there are more empty sets ?

In the hypothesis that there are more empty sets than the necessary one, a natural way to define an new empty set is to consider a function emptyset

emptyset : SETS --------> EMPTYSETS
emptyset (A) = A \ A

This is a well defined function. (this is the moment that I announce the existence of a logical science named 'zerology' - a read once an article some decades ago)

Suppose that A is [ 1, 2 } Then we can decide if 3, for example, belongs to emptyset(A). Well, 3 do not belongs to A, so it do not belongs to emptyset(A). In this manner, we can establish if every candidate is member of emptyset(A), that means the function emptyset is well defined.

A first property will be shown, after we will define

x ----- stands for intersection of sets
+ ----- stands for reunion of sets
< ----- stands for inclusion (or equality) of sets.
C ------ stands for the complementary of a set

FISRT THEOREM OF MANY EMPTY SETS
---------------------------------------------

for each sets A and B,

emptyset(A) x emptyset(B) < emptyset (AxB)

proof
------


Let x be in the left side member.

==== then ======
- x belongs to A
- x do not belongs to A
- x belongs to B
- x do not belongs to B
===== so ======
- x belongs to AxB
- x belongs to CA x CB that is included in CA + CB
===== so ======
- x belongs to AxB
- x belongs to C( AxB ), or x do not belongs to AxB
==== finally =========

x is in emptyset (AxB). Qvod erat demonstrandum.

That is why I said, do not enter this topic more for a smile Smile
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nick
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PostSubject: Re: Empty sets   Sun Sep 20, 2009 11:13 pm

Let me take a simpler example.

Let A an B be two sets, and A< B.

In this case, I can nothing deduce about the relation between emptyset(A) and emptyset(B).

This means that the previous message result is not so trivial as it looks.

===========

It makes fun for me, because I have just registered to three real analysis courses. I got the strange impression that we are 'squeezing' real numbers, in the same way one could 'squeeze' empty sets.

I feel like I watch ice flowers on a window behind it could be anything. I am watching only a projection of my own "logical glasses"...


===========

So, an element in emptyset(A) is a common element between A and nonA; it is strap, a bridge, or something that could be easily drawn on a Venn diagram.

These "jokers" that belong to A and also do not belong to A, could be noted like <a b> where a is in A and b is not in A.

For example, it the total world is { 1 2 3 4 5 } and A = { 1 2 } it is easy to see that emptyset (A) contains six straps
< 1 3 >, < 1 4 >, < 1 5 >, < 2 3 >, < 2 4 >, and < 2 5 >.

In this case, the model for an emptyset is an unoriented graph and the theory of empty sets is identical with the theory of graphs !

I think this is a good step ahead to define an algebra of emptysets even there are more models than the graphs one.
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bighouse
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PostSubject: ?   Fri Sep 25, 2009 1:15 am

What is this, magic ?
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nick
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PostSubject: Re: Empty sets   Fri Sep 25, 2009 10:24 pm

hello bighouse !

A kind of worm holes, like in Startrek, links between Universes Smile

There are many issues here,

- what about the TOTALSETS ?
- what if there is needed another map FI : EMPTYSETS ---> VERY EMPTYSETS etc
- could be constructed something coherent without loosing all that bunch of properties that a set algebra has ?

===============
by the way, I found something that use more copies of the empty set

http://www.geocities.com/complementarytheory/Successor.pdf

In this paper are relations like

{{},{},{},{}} is simpler than {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}
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nick
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PostSubject: Re: Empty sets   Fri Oct 16, 2009 11:36 am

The problem with the empty set is that the notation { } do not allow me to write more "copies of the empty set".

Let's note that there are possible several writings for other structures , e.g
{ a, { b } } is equivalent as inner structure with
{ a, { a, b } } and also with
{ a, { b, * } }

On the other hand, the nature offers a lot of examples of empty things, like empty boxes, bottles etc; for example, I use empty bags when I go shopping.

One should rectify the notation { } ; this could be useful in the Theory of Species, which suppose that there are identities like :
EMPTYSET + EMPTYSET = TWOEMPTYSETS.

Here, EMPTYSET is the thing that two empty boxes have in common, and
TWOEMPTYSETS is the thing that two couples of empty boxes have in common.

Yeah, it makes sense for me,

EMPTYBOX + EMPTYBOX = TWOEMPTYBOXES
(1, 0, 0, 0, 0,...) + (1, 0, 0, 0, 0,...) = (2, 0, 0, 0, 0,...)
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