Bruno,
I appreciate your grade-school teaching. We (I for one) can use it.
I still find that whatever you explain is an 'extract' of what can be
thought of a 'set' (a one representing a many).
Your 'powerset' is my example.
All those elements you put into { }s are the same as were the physical
objects to Aristotle in his 'total' - the SUM of which was always MORE than
the additives of those objects.
Relations!
The set is not an inordinate heap (correct me please, if I am off) of the
elements, the elements are in SOME relation to each other and the "set"-idea
of their ensemble, to *form* a SET.
You stop short at the naked elements *together*, as I see.
*They wear cloths and hold hands. Mortar is among them.*
Maybe your math-idea can tolerate any sequence and hiatus concerning to the
'set', and it still stays the same, as far as the *"math-idea you
need"*goes,  but if I go further (and you indicated that ANYTHING can
form a set)
the relations of the set-partners comes into play. Not only those which WE
choose for 'interesting' to such set, but ALL OF THEM influencing the
character of that *"ONE".*
*Just musing.*
**
John



On Tue, Jul 14, 2009 at 4:40 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

> Hi Kim, Marty, Johnathan, John, Mirek, and all...
>
> We were studying a bit of elementary set theory to prepare ourself to
> Cantor's theorem, and then Kleene's theorem, which are keys to a good
> understanding of the universal numbers, and to Church thesis, which are the
> keys of the seven steps.
> I intend to bring you to the comp enlightenment :)
>
> But first some revision. Read the following with attention!
>
> A set is a collection of things, which in general can themselves be
> anything. Its use consists in making a many into a one.
> If something, say x,  belongs to a set S, it is usually called "element" of
> S. We abbreviate this by (x \belongs-to S).
>
> Example:
>
> A = {1, 2, 56}. A is a set with three elements which are the numbers 1, 2
> and 56.
>
> We write:
>
> (1 \belongs-to {1, 2, 56}), or (1 \belongs-to A), or simply 1 \belongs-to
> A, when no confusions exist. The parentheses "(" and ")" are just delimiters
> for easing the reading. I write \belongs-to the relation "belongs to" to
> remind it is a mathematical symbol.
>
> B = {Kim, Marty, Russell, Bruno, George, Jurgen} is  a set with 5 elements
> which are supposed to be humans.
>
> C = {34, 54, Paul, {3, 4}}
>
> For this one, you may be in need of spectacles. In case of doubt, you can
> expand it a little bit:
>
> C = {    34,     54,    Paul,   {3, 4}     }
>
> You see that C is a sort of hybrid set which has 4 elements:
>
>    - the number 34
>    - the number 54
>    - the human person Paul
>    - the set {3, 4}
>
> Two key remarks:
> 1) the number 3 is NOT an element of C. Nor is the number 4 an element of
> C. 3 and 4 are elements of {3, 4}, which is an element of C. But, generally,
> elements of elements are not elements! It could happen that element of
> element are element, like in D = {3, 4, {3, 4}}, the number 3 is both an
> element of D and element of an element of D ({3, 4}), but this is a special
> circumstance due to the way D is defined.
> 2) How do I know that "Paul" is a human, and not a dog. How do I know that
> "Paul" does not refer just to the string "paul". Obvioulsy the expression
> "paul" is ambiguous, and will usually be understood only in some context.
> This will not been a problem because the context will be clear. Actually we
> will consider only set of numbers, or set of mathematical objects which have
> already been defined. Here I have use the person Paul just to remind that
> typically set can have as elements any object you can conceive.
>
> What is the set of even prime number strictly bigger than 2. Well, to solve
> this just recall that ALL prime numbers are odd, except 2. So this set is
> empty. The empty set { } is the set which has no elements. It plays the role
> of 0 in the world of sets.
>
>
> We have seen some *operations* defined on sets.
>
> We have seen INTERSECTION, and UNION.
>
> The intersection of the two sets S1 = {1, 2, 3} and S2 = {2, 3, 7, 8} will
> be written (S1 \inter S2), and is equal to the set of elements which belongs
> to both S1 and S2. We have
>
> (S1 \inter S2) = {2, 3}
>
> We can define (S1 \inter S2) = {x such-that ((x belongs-to S1) and (x
> belongs-to S2))}
>
> 2 belongs to (S1 \inter S2) because ((2 belongs-to S1) and (2 belongs-to
> S2))
> 8 does not belongs to (S1 \inter S2) because it is false that ((2
> belongs-to S1) and (2 belongs-to S2)). Indeed 8 does not belong to S1.
>
> Of course some sets can be disjoint, that is, can have an empty
> intersection:
>
> {1, 2, 3} \inter {4, 5, 6} = { }.
>
> Similarly we can define (S1 \union S2) by the set of the elements belonging
> to S1 or belonging to S2:
>
> (S1 \union S2) = {x such-that ((x belongs-to S1) or (x belongs-to S2))}
>
> We have, with S1 and S2 the same as above (S1 = {1, 2, 3} and S2 = {2, 3,
> 7, 8}):
>
> (S1 \union S2) = {1, 2, 3, 7, 8}.
>
> OK. I suggest you reread the preceding post, and let me know in case you
> have a problem.
>
>
> We have seen also a key *relation* defined on sets: the relation of
> inclusion.
>
> We say that (A \included-in B) is true when all elements of A are also
> elements of B.
>
> Example:
> The set of ferocious dogs is included in the set of ferocious animals.
> The set of even numbers is included in the set of natural numbers.
> The set {2, 6, 8} is included in the set {2, 3, 4, 5, 6, 7, 8}
> The set {2, 6, 8} is NOT included in the set  {2, 3, 4, 5, 7, 8}.
>
> When a set A is included in a set B, A is called a subset of B.
>
> We were interested in looking to all subsets of a some set.
>
> What are the subsets of {a, b} ?
>
> They are { }, {a}, {b}, {a, b}. Why?
>
> {a, b} is included in {a, b}. This is obvious. All elements of {a,b} are
> elements of {a, b}.
> {a} is included in {a, b}, because all elements of {a} are elements of {a,
> b}
> The same for {b}.
> You see that to verify that a set with n elements is a subset of some set,
> you have to make n verifications.
> So, to see that the empty set is a subset of some set, you have to verify 0
> things. So the empty set is a subset of any set.
>
> proposition: { } is included-in any set.
>
> So the subsets of {a, b} are { }, {a}, {b}, {a, b}.
>
> But set have been invented to make a ONE from a MANY, and it is natural to
> consider THE set of all subsets of a set. It is called the powerset of that
> set.
>
> So the powerset of {a, b} is THE set {{ }, {a}, {b}, {a, b}}. OK?
>
> Train yourself on the following exercises:
>
> What is the powerset of { }
> What is the powerset of {a}
> What is the powerset of {a, b, c}
>
> Any question?
>
> This was a bit of revision, to let Kim catch up.
>
> The sequel will appear asap. Be sure everything is OK, and please, ask
> question if it is not.
> You can also ask any question on the first sixth steps of UDA ('course).
>
> Bruno
>
>   http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
>

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