Saturday, October 13, 2012

24. Domain of Relations and their Converse

 
Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.
 
Similarly the class of all terms to which something or other has the relation R is called the converse domain of R; it is the same as the domain of the converse of R. The converse domain of R is represented by "ʻR"; thus

ʻR = y^{(x) . xRy}  Df.

The sum of the domain and the converse domain is called the field, and is represented by CʻR: thus

CʻR = DʻR ʻR   Df.  

The field is chiefly important in connection with series. If R is the ordering relation of a series, CʻR will be the class of terms of the series, DʻR will be all  the terms except the last (if any), and ʻR will be all the terms except the first (if any). The first term, if it exists, is the only member of DʻR -ʻR, since it is the only term which is a predecessor but not a follower. Similarly the last term  (if any) is the only member of ʻR DʻR. The condition that a series should have no end is ʻR DʻR, i.e. "every follower is a predecessor"; the condition for no beginning is DʻR ʻR. These conditions are equivalent respectively to DʻR = CʻR  and ʻR = C'R.

The relative product of two relations R and S is the relation which holds between x and z when there is an intermediate term y such that x has the relation R to y and y has the relation S to z. The relative product of R and S is represented by R|S; thus we put

                     R|S  =  x^y^{(y) . xRy. ySz}  Df,

whence:       : (R|S) z . ≡ . (y). xRy. ySz.

Thus "paternal aunt" is the relative product of sister and father; "paternal grandmother" is the relative product of mother and father; "maternal grandfather" is the relative product of father and mother. The relative product is not commutative, but it obeys the associative law, i.e.

. (P|Q)|R  = P|(Q|R).

It also obeys the distributive law with regard to the logical addition of relations, i.e. we have.

 . P|(Q R) = (P|Q) (P|R),

. (Q R)|P  = (Q|P) (R|P).

But with regard to the logical product, we have only

. P| (Q R) (P|Q) (P|R),

.(Q R)|P (Q|P)   (Q|R).  

The relative product does not obey the law of tautology, i.e. we do not have in general R  R = R. We put

R2=R|R  Df.

Thus paternal grandfather = (father)2,

maternal grandmother = (mother)2.

A relation is called transitive when R2 R, i.e. when, if xRy and yRz, we always have xRz, i.e. when

xRy . yRz . Éx,y,z. xRz.  

Relations which generate series are always transitive; thus e.g.

x > y . > z Éx,y,z . x > z.

If P is a relation which generates a series, P  may conveniently be read "precedes"; thus "xPy. yPz .x,y,z . xPz" becomes "if x precedes y and y precedes z, then x always precedes z." The class of relations which generate series are partially characterized by the fact that they are transitive and asymmetrical, and never relate a term to itself.

If P is a relation which generates a series, and if we have not merely P 2  P, but P2 = P, then P generates a series which is compact (überall dicht), i.e. such that there are terms between any two. For in this case we have

xPz. . (y). xPy. yPz,  

i.e. if x precedes z, there is a term y such that x precedes y and y precedes z,

i.e. there is a term between x and z. Thus among relations which generate series, those which generate compact series are those for which P2 = P.

Many relations which do not generate series are transitive, for example, identity, or the relation of inclusion between classes. Such cases arise when the relations are not asymmetrical. Relations which are transitive and symmetrical are an important class: they may be regarded as consisting in the possession of some common property.
By Hy Mayer circa 1901 © Respective copyright holders.
 
 
 
 
 
 
 
 
 
Looking deeper into denoting relations and their interactions we meet the dizzy complexity that this system turns into controlled order and exclusion of error in equations that take us where we hope to be with our understanding and expression.
Krazy Kat and Ignatz by The Venerable George Herriman
Although I am writing this blog for my own study and search for clarity, there exists a disjunction of intention that I cannot reconcile; nor do I find it troubling, or necessary to do so. On one hand I see this work on Principia Mathematica as the path that my life has led me to master on my own terms. Then also I see that it is only a means to turnaround and accomplish my work in progress now: Sweet Neurology of My Heart on human brain chemistry and the subconscious mind; that, at last, expresses the sum total of my knowledge gained in 45 years as a working artist.
 
As a young man I started out to remain invisible, to create art on my own terms, detached from mundane survival by means of alternate skills with the goal of, having my time to be as I choose, and to live in it, and create art, study, socialize and enjoy  my life in a mystic urban city that has bonded with my invisible nature. Always I have shared my artwork, my written work, mentoring, teaching, and all forms of shaman healing individually. If a dozen people read one of my books or poems in a given year, if a hundred people saw my paintings or murals or installations, that was entirely satisfactory to me.
 
The disjunction in me now arises from this blog. I love making this artwork, it is the fulfillment of a craving that has driven me since I was a boy; to express the aggregate of my  thought as a cohesive and encompassing whole. I am gratified by the modest traffic to this blog. I wonder if it is intelligible to readers who stumble upon me looking for Indians, but who find 700 word posts full of equations. To me it reads like a ‘Tumbleweeds’ comic strip, but it would not have 6 months ago. I have always prized any criticism of my work, but this blog hasn’t drawn a peep. I know I am paddling against the online cultural current. Other blogs are read in seconds while I ask for long and extended snatches of concentration. I have never been one to follow trends. My goal is to build healthy and extended attention spans in persons who choose to read my blog. One to me is many. You a reader of my work are everything.
  Script: Gerry Conway  Pencils and inks: Patton/Machlan   Flash™ Superman™ WonderWoman™                                     © Respective copyright/trademark holders.


Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.

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