On 08 Jul 2009, at 15:43, m.a. wrote: > Second try: >> (power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}, {1,2}, {2,3}, {1,2,3}}

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This is far better! Not yet correct though. I gave you the hint that there are 8 elements. Let us count: The empty set { } ..................................1 Three singletons {1}, {2}, {3}................3 Two doubletons {1,2 }, {2,3 }................2 The biggest subset {1,2,3}..................1 1 + 3 + 2 + 1 = 7 A subset is missing! Can you see which one? > And I give you a little subject research: if a set x has n elements, > how many elements are in (power x)? > > How's this for a wild guess? I have a feeling that it's missing the > accolades, but I have no idea where to put them > > (power {x}) = n(n-1) (n-2)...(n-x+1) > x! > Good intuition that there is a problem with the accolades. Although your expression is not *missing* accolades. Actually it has *too much* accolades. If x represents a set, for example the set {1,2}, it means that in the formula x can be substituted by {1,2}. we could write that x = {1, 2}, the accolades are in x, if you want. So on the left, you should have written (power x), like in the enunciation of the subject research, actually. If x is the set {1,2}, (power x) is (power {1, 2}). But (power {x}) is (power {{1,2,3}}), i.e. the powerset of {{1,2,3}}, which is {{ }, {{1,2,3}}}. (Power {x}) looks like the powerset of an indeterminate singleton, a set with only one element. You could have written this: When x has n elements, then (power x) = n(n-1) ... (n-x+1) / x! Let us see. After all you already computes the powerset of { }, which is the set with 0 element, and you told me (power { }) = {{ }}. So it has one element, and your formula should confirm this, and ... well, your formula begins by n multiplied by something, if n = 0 then we will get 0, because 0 times any number gives 0. But we have just seen that (power { }) = {{ }}, which is a set with 1 element. So your formula is already contradicted by the first example. Hmm... May be that was bad luck, and sometimes in math the first example is also the trickiest, so let us look for n = 1. Let us take a set with one element, like {24}. Its power has 2 elements: {{ } {24}}, and you can guess that all singletons (set with two elements) have the same number of elements in their power set. So the answer is 2, in this case. If x has 1 element, the powerset of x, (power x) has two elements. Your formula should give 2, when n is equal to 1. Let us see ... it gives 1(1-1) ... (1 - ... but now, what could you mean by (n - x ...). ??? n is supposed to represent a number, x is supposed to represent a set, how could I, or you, subtract a set from a number? So I'm afraid that your formula is senseless, although I will perhaps take the time, in some future, to explain why there *is* an atom of truth in it! The correct formula is much simpler, though. Morality: if you have a theory, or a formula, test it on what you already know before submitting to publication! To find the formula, you could try first the tedious brut force (if this is english). (Few mathematicians admit to do that, but all mathematicians do it!) The number of sets included in { } = 1 (you have seen that). If x has 0 elements, (power x) = 1. The number of sets included in {a} = 2 (you have seen that). If x has 1 elements, (power x) = 2. The number of sets included in {a, b} = 4 (you have seen that). If x has 2 elements (power x) = 4. The number of sets included in {a,b,c} = 8 (cf the hint, exercise above). If x has 3 elements (power x) = 8 Could you compute and/or guess the number of sets included in {a,b,c,d} ? And what about {a,b,c,d,e}?, and {a,b,c,d,e,f} ?, and ... ? Bruno PS Did I say that (power x) is called the powerset of x? It could be better to write (powerset x) instead of (power x), to emphasize that the powerset of x is the *set* of the sets included in x. OK? (Sorry). 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 -~----------~----~----~----~------~----~------~--~---