2011-06-28

classification by superset decomposition

5.15: adda/oop/type cluster/
templates for subtype association:

[6.1: mis:
"( . having types specifying which subset relations exist
doesn't neatly show the relation of irrationals
)??
irrationals are simply the reason
why some reals are not Q's !
. what is the purpose of knowing how they are related?
isn't it just to know type compatibility?
if you produce an irrational,
you call it a real,
and if you didn't want symbolics
then you'd call it a float .]

. the mgt for type clusters(eg numbers),
may have some reusable code in that there are
many type clusters in which the subtypes have
values that are subsets of each other; eg,
N subsets Z subsets Q subsets R subsets C .

. number is also an example of there being
another way of describing subsets:
superset decomposition
(showing how a set is composed of other sets).
. saying which subset relations exist
doesn't neatly show the relation of irrationals
(ones not expressible as a ratio, and
having an infinite and non-recurring expansion
when expressed as a decimal).

N = 0.. infinity;
Z= {+1,-1}*N;
Q = {Z/N, Z};

irrationals = {pi, e, 2**(1/2), ...}:
pi/4 = +(^i=0...infinity| (-1)**i /(2i+1) )
pi/2 = *(^i=1...infinity| (2i)**2 /(2i)**2-1) )
pi = 4 / (1+ 1**2 /( 3+ 2**2 /(5+ 3**2 /(...)))
e = (1+1/infinity)**infinity .

R = {Q, irrationals, repeaters}
C = {i*R + R, R} -- i = (-1)**(1/2)