
Re: set builder notation
Posted:
Aug 17, 2013 12:30 PM


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}.
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?

