Re: randomness


Lets say that a + ~b and c + ~d are a pair of partial randoms. Let b,d, be
sets of discrete values {b1,b2,..bn}, {d1,d2,..dn}. a and c are reals.

Also define e + ~f and g + ~h, where e,g are real and ~f represents a
random chosen from the continuous interval [f1, f2], likewise ~h is a random
from the interval [h1, h2].

s and t are just real numbers

It sure seems reasonable that
(a + ~b) + (c + ~d) = (a+c) + ~{b union d}

and, that
( e + ~f ) + ( g + ~h ) = ( e + g ) + ~[f1 + h1, f2 + h2]

getting wierder .........

(a + ~b) + (c + ~d) = (c + ~d) + (a + ~b)

and

( e + ~f ) + ( g + ~h ) = ( g + ~h ) + ( e + ~f )

We might even attempt multiplication, something like this does seem to make
some sense intuitively, so maybe it's not complete garbage :

s * (a + ~b) = s * a + s * ~b = s*a + s*{b1,b2,..bn} = s*a + {s*b1,
s*b2,.....s*bn}

......just thinking about this.....it might even make sense.....so........

t * ( e + ~f ) = t*e + t* ~f = t*e + t*~[f1, f2] = t*e + ~[t*f1, t*f2]

And you know something, I think that we are getting close to having a ring
or a field or something, who knows / who cares, it will all blow up
anyway......


Does it make sense that there uis an operation
(a + ~b) * (a + ~b) ????

What about
( e + ~f ) * ( e + ~f ) ???????


What the heck is {1,2,3,4,5,6} ^2 ?????

What about [0,5] ^ 2 ?????

How to interpret these things ???

Cartesian geometry ? Wierd.










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Definition of a Field
Both + and * are associative
For all a, b, c in F, a + (b + c) = (a + b) + c and a * (b * c) = (a * b) *
c.
Both + and * are commutative
For all a, b belonging to F, a + b = b + a and a * b = b * a.
The operation * is distributive over the operation +
For all a, b, c, belonging to F, a * (b + c) = (a * b) + (a * c).
Existence of an additive identity
There exists an element 0 in F, such that for all a belonging to F, a + 0 =
a.
Existence of a multiplicative identity
There exists an element 1 in F different from 0, such that for all a
belonging to F, a * 1 = a.
Existence of additive inverses
For every a belonging to F, there exists an element ?a in F, such that a +
(?a) = 0.
Existence of multiplicative inverses
For every a ? 0 belonging to F, there exists an element a?1 in F, such that
a * a?1 = 1.


A Ring is a set R equipped with two binary operations + : R × R ? R and · :
R × R ? R, called addition and multiplication, such that:
(R, +) is an abelian group with identity element 0:
(a + b) + c = a + (b + c)
0 + a = a + 0 = a
a + b = b + a
For every a in R, there exists an element denoted ?a, such that a + ?a = ?a
+ a = 0
(R, ·) is a monoid with identity element 1:
(a·b)·c = a·(b·c)
1·a = a·1 = a
Multiplication distributes over addition:
a·(b + c) = (a·b) + (a·c)
(a + b)·c = (a·c) + (b·c)






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