On 07 Jul 2009, at 16:18, m.a. wrote:

> Thanks, Bruno. I think I've got it now. Sorry to be such a slow  
> learner.
>                                                                               
>                                      marty


Slow? It seems to me that you are rather quick. A scientist friend on  
mine took 4 years to get the difference between "belongs-to" and  
"included-in".
Of course it will take more time you get friendly familiar with such  
matter, but that is completely normal.
You are not slow at all, and I appreciate your seriousness and your  
courage, because you can say publicly "I don't understand, explain  
again". It is the big virtue which will help you to proceed.

Now, understanding is but one phase. You will have to remember what  
you learn. For this I suggest you do by yourself some summaries, and  
as I said it could help to try to explain to some others. But there is  
no problem to ask any question, or to suggest me to recall definitions  
and examples.
At some point I will sum up myself.

Oh, let me sum up a few bit before I introduce a new notion, and three  
exercise (but you can take some holiday before, take your time).

---

What are sets? Sets are sort of boxes which can contains anything,  
like numbers, or sets. Most of the set we have encounter were set of  
numbers, or set of sets.

You can perhaps intuit some use of set in logic. For example saying  
that being a human makes you mortal can be analysed by the statement  
that the set of humans is included in the set of mortal beings. The  
proposition "Julia is a human" is equivalent with the proposition that  
Julia belongs to the set of humans. If I let H be a name for the set  
of humans, M be the name of the set of mortal beings, and j be a name  
for Julia, the fact that Julia is human, can be translated in "set  
theory" by

(j belongs-to H),

and the fact that being a human makes you mortal, can be translated by

(H included-in M);

Remember that (H included-in M) means that all elements of H are  
element of  M, and so it means that if j belongs-to H then j belongs- 
to M.

A logician would say that with the axioms (j belongs-to H) and (H  
included-in M), you can deduce that (j belongs-to M).

A logician never care if the axioms are true or false, he cares only  
on the validity of the reasoning.

Remark. Personally, I don't believe that in "real life" there are  
sets, like those we can meet in math. Take the set of humans. Do we  
have a really a set ? An anti-computationalist could classify Julia as  
an inhuman zombie the day she got her artificial brain, so H is  
already different for a computationalist and an anti- 
computationalist!  In real life, sets can be locally useful, but it  
would be a sort of occamization, to quote John, (inspired by Russell)  
to apply the notion of set so straightforwardly. I have the same  
opinion for the use of set in mathematics, concerning the long run,  
but then I understand how much they make thinks far easier and  
clearer. Indeed they pervade all branches of math: topology,  
probability, algebra, logic, and computer science is no exception.  (I  
think they will disappear, but this will take millenia!)

------

Now it is time to do some exercise.

Do you remember, I asked you to give me all the subsets of {1, 2}.  
That is, all the sets which are included in {1, 2}. You gave me the  
correct answer: those subsets are { }, {1}, {2}, {1, 2}. You see that  
the set {1, 2} has 2 elements, and 4 subsets. But then I asked to give  
me the set of all subsets of {1, 2}.
{1, 2} has four subsets, and it is natural to make that many a one, by  
considering *the* set of all subsets of {1, 2}. The answer is:

{{ }, {1}, {2}, {1, 2}}

Considering all subsets of a set is a rather important operation,  
which we will meet more than one times in the sequel. Given its  
importance mathematicians gave it a name. It is the power operation.  
Later I will be able to explain why it is called power.
It is an UNARY operation, which means it applies on ONE set.  
(Intersection, and union are BINARY operations, they need two sets to  
work on).

So (power x) = {y such-that y is included in x}, by definition.

For example:
(power {1, 2}) = {{ }, {1}, {2}, {1, 2}}

Here are the three promised exercises. Compute


(power {1}) = ?
(power {1, 2, 3}) = ?
(power { }) = ?

Take your time,

Bruno




http://iridia.ulb.ac.be/~marchal/




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