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Topic: PREDICATE CALCULUS 2
Replies: 23   Last Post: Nov 29, 2012 1:13 AM

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Graham Cooper

Posts: 4,237
Registered: 5/20/10
Re: PREDICATE CALCULUS 2
Posted: Nov 27, 2012 8:59 PM
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There are 2 ALLs which is more complicated but you can format it as a
SUBSET using a cartesian product of the 2 X values with a common Y.


isfunction(r) <- ALL(Y1) ALL(Y2) r(X,Y1)^r(X,Y2) -> Y1=Y2

{(Y1,Y2)|r(X,Y1)^r(X,Y2)} C {(Y1,Y2)|r(X,Y1)^r(X,Y2)->Y1=Y2}

...


TEST IF IT CAN DETECT FUNCTIONS!

-----r--------X---Y
proposed( tom, jane )
proposed( mark, sally)
proposed( josh, sally )
proposed( henry, barb )


{ (Y1,Y2) | r(X,Y1)^r(X,Y2) }
= {
(jane,jane) *** X= tom
(sally,sally) *** X = mark
(sally,sally) *** X = josh
(barb,bar) *** X = henry
}

{ (Y1,Y2) | r(X,Y1)^r(X,Y2)->Y1=Y2 }
= {
(jane,jane) *** X= tom
(sally,sally) *** X = mark
(sally,sally) *** X = josh
(barb,bar) *** X = henry
}

L.H.Set C R.H.Set
'proposed' is a function, only 1 proposal per man (X).

-----------------------------

OK try the reverse function

----r------X1---X2
kisssed( jane, tom )
kisssed( sally, mark)
kisssed( sally, josh )
kisssed( barb, henry )


{ (Y1,Y2) | r(X,Y1)^r(X,Y2) }
= {
(tom,tom) *** X= jane
(mark,mark) *** X = sally
(josh,josh) *** X = sally
(mark,josh) *** X = sally
(josh,mark) *** X = sally
(henry,henry) *** X = barb
}

{ (Y1,Y2) | r(X,Y1)^r(X,Y2)->Y1=Y2 }
= {
(tom,tom) *** X= jane
(mark,mark) *** X = sally
(josh,josh) *** X = sally
(henry,henry) *** X = barb
}


L.H.Set is NOT C R.H.Set

so KISSED is not a function.

Herc
--

{(Y1,Y2)|r(X,Y1)^r(X,Y2)} C {(Y1,Y2)|r(X,Y1)^r(X,Y2)->Y1=Y2}
DETECTING r is a FUNCTION without QUANTIFIERS!




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