fom
Posts:
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Registered:
12/4/12


Re: set builder notation
Posted:
Aug 17, 2013 3:20 PM


On 8/17/2013 6:12 AM, 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? > > The vertical bar is essentially conjunction, correct? >
Yes. But this is a special form.
In logical notation, this is a restricted quantification,
All x ( x in A /\ P(x) )
Exists x ( x in A > P(x) )
Note that a restricted universal quantifier uses a conjunction while a restricted existential quantifier uses a conditional.
Arbitrary use of class abstraction operators in the form of
S = {x  P(x)}
correspond with unrestricted quantification
All x ( P(x) )
Exists x ( P(x) )
and can lead to the paradoxes of set theory.
Restricted quantification is sometimes written as
All [x in A] ( P(x) )
Exists [x in A] ( P(x) )
This masks the difference between the quantifiers made explicit by the use of connectives as above.
Herc has observed that the use of restricted quantification changes the logic from the familiar firstorder logic to a higherorder logic. In order to retain certain familiar logical results, the entire language (that is, the P(x)'s) must be built up with restricted quantifiers in accordance with restrictions based upon negation normal forms.

