John Randall wrote, in the Programming forum: > I think the author of the Wikipedia article is trying to get at this, > so you would accept 1+1=2 (mod 12) but not 9+4=1 (mod 12). In my > opinion, this fixates on the distinguished representatives rather > than the equivalence classes, and if you do that, you will go wrong > somewhere else.
Thank you for this assessment of that flawed Wikipedia paragraph. Because equivalence classes are infinite sets, my curiosity is piqued: How should Iverson notation be applied when dealing with infinite sets? It seems that for this use we must move to statements that would not be valid under the requirements of executability that J and APL entail. Yet, it would be nice to be able to refer to things like these equivalence classes, integers, rational numbers, real numbers, and the complex plane within a near-J notational structure. I've put this to the Chat forum because it departs from J proper, but if the moderators think it is a better fit for the Programming forum we can move it back there. Another area where I'm not entirely sure how to apply J are statements of formal logic such as "for all" and "there exists" (inverted capitals A and E, respectively.) In general, I wish to explore a return to the roots of J. The abstraction of tacit notation is itself a triumph along those lines; the way it removes specification of particulars allows us to refer strictly to the functions. The arguments could in many cases be infinite sets, setting aside the needs of implementation. The limit power (^:_) also provides something along these lines. As an example of where my curiosity has wandered, I've not been able to decide whether (i: _j_) naturally denotes the Rationals, or only a subset of them. (If "_" were taken as a value, which it must not be, it would denote the Integers.) -- Tracy Harms ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
