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}}

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/




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