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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

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.
>>>
>>> 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

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

Date Subject Author
8/17/13 lite.on.beta@gmail.com
8/17/13 Graham Cooper
8/24/13 lite.on.beta@gmail.com
8/17/13 Peter Percival
8/17/13 LudovicoVan
8/17/13 David C. Ullrich
8/17/13 Peter Percival
8/18/13 David C. Ullrich
8/18/13 fom
8/18/13 Shmuel (Seymour J.) Metz
8/18/13 fom
8/19/13 Graham Cooper
8/17/13 fom