Torgny Tholerus skrev:
>
> Exercise: Show that the extended Leibniz rule in the discrete
> mathematics: D(f*g) = f*D(g) + D(f)*g + D(f)*D(g), is correct!
>
Another way to see both form of the Leibniz rule is in the graphical set
theory, where you represent the sets by circles on a paper. Here I will
represent the union of the sets A and B with "A + B", and the
intersection as "A*B".
Then you can represent the D operator as the border of the circle.
Then you will have:
D(A*B) = A*D(B) + D(A)*B, ie the Leibniz rule, ie the border of the area
of the intersection is the union of the border of B inside A, and the
border of A inside B. I can not show this figure in this message, but
you can draw two circles on a paper before you, and you will then see
what I mean.
Now the interesting thing is what will happen if the circles have
*thick* borders: Then the set A is represented by two circles inside
each other, and the border will then be the area between the two
circles. The set A will then be the interior of the inner circle, and
the outside of A will be the outside of the outer circle.
What will you then get if you look at the border of the intersection of
A and B?
This time you will get:
D(A*B) = A*D(B) + D(A)*B + D(A)*D(B), ie the extended Leibniz rule. The
extra term then comes from the two small squares you get where the two
borders cross each other. (Do draw this figure om the paper before you,
and you will understand.)
This picture with the circles with thick borders is a way to represent
intiutionistic logic. The interior of the inner circle is the objects
that represent A (such as "red"), and the outside of the outer circle
represent not-A (such as "not red"). Inside the border you will have
all that is neither A nor not-A (such as red-orange, where you don't
know if it is red or not...)
--
Torgny Tholerus
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Received on Fri Nov 14 2008 - 11:03:09 PST