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

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Peter Percival

Posts: 1,302
Registered: 10/25/10
Re: set builder notation
Posted: Aug 17, 2013 1:47 PM
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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)'? 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)}.

>
> Then S an element of S implies S not an element
> pf S, and conversely; there is no such set S.
>
> Mathhematians other than set theorists use
> (3) all the time, but officially it has to be (1).
>
>

>>
>> The vertical bar is essentially conjunction, correct?

>


--
Sorrow in all lands, and grievous omens.
Great anger in the dragon of the hills,
And silent now the earth's green oracles
That will not speak again of innocence.
David Sutton -- Geomancies



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