
Re: set builder notation
Posted:
Aug 18, 2013 10:03 AM


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)".
> >> >> 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? >>

