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Topic: set builder notation
Replies: 12   Last Post: Aug 24, 2013 1:38 AM

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fom

Posts: 1,968
Registered: 12/4/12
Re: set builder notation
Posted: Aug 18, 2013 12:36 PM
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On 8/18/2013 9:03 AM, dullrich@sprynet.com wrote:
> On Sat, 17 Aug 2013 18:47:40 +0100, Peter Percival
> <peterxpercival@hotmail.com> wrote:
>

>> dullrich@sprynet.com wrote:
>>> On Sat, 17 Aug 2013 04:12:34 -0700 (PDT), lite.on.beta@gmail.com
>>> wrote:
>>>

>>>>
>>>> S = {x /in A | P(x) }
>>>>
>>>> For the set builder notation above, what we really means is:
>>>>
>>>> all things x, such that "x is element of A *and* P(x) is true" correct?

>>>
>>> Yes and no.
>>>
>>> Yes:
>>>
>>> (1) {x in A | P(x)}
>>>
>>> is the same as
>>>
>>> (2) {x | x in A and P(x)}.
>>>
>>> No:
>>>
>>> No, because (2) is actually not a "legal"
>>> construction of a set! (2) is of the form
>>>
>>> (3) {x | Q(x)},
>>>
>>> and things of the form (3) are officially not
>>> allowed.
>>>
>>> Not allowed because they lead to contradictions:
>>> Let
>>>
>>> S = {x | x is not an element of x}.

>>
>> But is 'x is not an element of x' of the form 'x in A and P(x)'?

>
> No.
>

>> I
>> suppose what I'm doing is challenging you to reproduce Russell's paradox
>> with sets of the form {x | x in A and P(x)}. I'm rather sure (but I
>> know nothing) that {x in A | P(x)} is *nothing but* alternative notation
>> for {x | x in A and P(x)}.

>
> Informally it's just an alternate notation.
>
> The fact that {x | P(x)} is not allowed seems more important.
> If you want you can change things so that {x | P(x)| is
> allowed as long as P(x) is of the form "x in A and Q(x)".
>


Yes.

One can find a discussion of a language in this form
in "Topological Model Theory" by Flum and Zeigler

http://books.google.com/books/about/Topological_Model_Theory.html?id=YpkfAQAAIAAJ

The introduction of restricted quantifiers via the
formation rules are also constrained relative to
equivalences with negation normal forms.






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