Marty,

On 08 Jul 2009, at 02:28, m.a. wrote:

>
> ----- Original Message -----
> From: Bruno Marchal
> To: everything-list@googlegroups.com
> Sent: Tuesday, July 07, 2009 2:07 PM
> Subject: Re: The seven step series
>
>
> On 07 Jul 2009, at 16:18, m.a. wrote:
>
>>
> Bruno,
>
> I'm not entirely sure of these answers, but I think I learn more  
> from your corrections than from pondering the rules to the point  
> where I confuse myself.   m.a.

Hmm... You are not slow, but you may be a bit lazy :)
Pondering the rules to the point where you confuse yourself is  
necessary to develop ... well, the art of pondering the rules to the  
point where you confuse yourself, which is part of the work of the  
researcher.
But that is OK Marty, given that the goal here is just to give the  
necessary passive understanding of math so as making you able to grasp  
the seventh step of UDA.


>
> 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}) = ? {{ }, {1}}

Excellent.


> (power { }) = ?   {{ }}

Excellent. People are often wrong on this one!


> (power {1, 2, 3}) = ? {{ }, {1}, {2}, {3}}


Here you are a bit lazy, as I said. You miss many subsets. Is not {1,  
2} a subset of {1, 2, 3}? Is not {2, 3} a subset, and where is {1, 2,  
3}?

I have to go right now, so I let you search, meanwhile, for the  
complete solution by yourself. I give you a hint (power {1,2,3}) has 8  
elements.

And I give you a little subject research: if a set x has n elements,  
how many elements are in (power x)?

Bruno

http://iridia.ulb.ac.be/~marchal/




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