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

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fom

Posts: 1,968
Registered: 12/4/12
Re: set builder notation
Posted: Aug 17, 2013 3:20 PM
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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 first-order logic
to a higher-order 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.




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