[Arisbe] Abstraction, Analogy, Example, Icon, Metaphor, Model, Morphism, Paradigm, Prototype, Simulation

[Jon Awbrey]

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AAEIMMMPPS.  Note 2

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Copy of a Note at PlanetMath --

Subj: "SymDif = Plus" by Jon Awbrey on 2008-05-02
Note: http://planetmath.org/?op=getmsg&id=18843

There's a discussion in this old application paper of mine:

http://www.mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems#Special_Classes_of_Propositions

Let B = {0, 1}

Let x_i for i = 1 to n be the characteristic functions
of the sets A_i, i = 1 to n, all subsets of universe X.

Consider the "factoring" of a proposition f : X -> B,
where f is 1 of the 2^(2^n) functions based on the x_i,
as the composite function f*(c(x)), where c : X -> B^n
is the "coding" of each x in X as an n-bit string in B^n,
and where f* is the mapping of these code n-tuples into B.

............ X ... f ... B
............. o-------->o
.............. \ ..... ^
{x_1, ..., x_n) \ ... / f*
................ \ . /
................. v /
.................. o
................. B^n

There are 2^n linear functions h : B^n -> B,
with (n choose k) linear functions of rank k,
for k = 0 to n.

These linear functions are the same things as the
symmetric differences of rank k, for k = 0 to n.

A typical one can be written as a k-fold sum,
x_i_1 + x_1_2 + ... x_i_(k-1) + x_i_k, where
"+" signifies the GF(2) addition, also known
as exclusive disjunction (XOR) or Sym.Dif.

I hope I got this right -- it's been a while.

Jon Awbrey

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