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

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David C. Ullrich

Posts: 3,164
Registered: 12/13/04
Re: set builder notation
Posted: Aug 17, 2013 12:30 PM
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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?





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