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/ > > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---