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/ --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---