| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 1214.1 | | AITG::DERAMO | Dan D'Eramo, nice person | Tue Mar 27 1990 17:17 | 17 |
| Use the equivalences: not for all x not <==> the exists an x
not there exists a y not <==> for all y
not not <==>
So
for all x, there exists a y such that P(x,y)
<==>
not there exists an x not not for all y not P(x,y)
<==>
not there exists an x for all y not P(x,y)
So the negation of that will be just
there exists an x such that for all y, not P(x,y)
Dan
|
| 1214.2 | | JARETH::EDP | Always mount a scratch monkey. | Tue Mar 27 1990 17:18 | 13 |
| Re .0:
I don't think you can simply invert the quantifiers. Why do you want
to? If you just want "there exists" to appear for "for all", you can
write the statement as:
- (there exists an x such that (for all y, -P(x,y))).
Note that the first possibility you wrote is actually the negation of
the original statement; the statement above is equivalent to the
original.
-- edp
|
| 1214.3 | | AITG::DERAMO | Dan D'Eramo, nice person | Tue Mar 27 1990 17:25 | 17 |
| Essentially, a not can trade places with a for_all or a
there_exists if in doing so it flips the quantifier to
the other kind.
not for_all x there_exist y p(x,y)
not for_all x there_exist y p(x,y)
not for_all x there_exist y p(x,y)
not for_all x there_exist y p(x,y)
not for_all x there_exist y p(x,y)
B U M P !
there_exists x not there_exist y p(x,y)
B U M P !
there_exists x for_all y not P(x,y)
The running start helps but isn't necessary. :-)
Dan
|