On Tue, Jul 10, 2007 at 02:39:28PM +0200, Sebastian Haase wrote: > When you talk about algebra - one might have to restrict one self to '|' and > '&' > -- not use '+' and '-' > E.g.: > True - True = False # right !? > # but if: > True+True = True. > # then > True+True -False = True -False # ???? > # here I'm already lost ... I don't think this can be done in a consistent > way.
It can, its called the Bool algebra, and it is a consistent algebra, in a mathematical sense of algebra (http://en.wikipedia.org/wiki/Boolean_algebra), actually what we are talking about is the two element bool algebra (http://en.wikipedia.org/wiki/Two-element_Boolean_algebra), and the mathematical structure we are taling about is a ring, the wikipedia article is quite comprehensible (http://en.wikipedia.org/wiki/Ring_(mathematics)) > In other words: a "+" operator would also need a corresponding "-" Yes. In other words (the ensemble theory words) each element needs to have an opposite concerning the '+' law. To understand this you need a bit of algebra theory. * An algebra has 2 laws, lets call them "+" and "*". * Each law has a neutral element for this law, ie an element a for which "a + b = b" for all b in the algebra, lets write these "n+", and "n*". * Each element a is required to have an inverse for the "+", ie an element b for wich b + a = n+, lets write the opposite of b "-b". For integer, n+ = 0, n* = 1. For Booleans, n+ = False, and n+ = True, therefore, as Matthieu points out, -True = True, as True + True = n+ = True, and -False = True, as True + False = n+ = True. So you have a consistent algebra. Now there is a law for which every element does not have an inverse, it the "*" law. You can check the out for integers. It is also true for booleans. In fact, you can proove that in an ring, n+ cannot have an inverse for the * law (it the famous divide by zero error !). In conclusion, I would like to stress that, yes, +, - and * are well defined on booleans, the definition is universal, and please don't try to change it. Gaël _______________________________________________ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
