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Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark
holders.
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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.
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By Hy Mayer circa 1901 © Respective copyright
holders.
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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.
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| 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.
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Script: Gerry Conway Pencils and inks: Patton/Machlan Flash™ Superman™ WonderWoman™ © Respective
copyright/trademark holders.
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Harold R. (Hal)
Foster’s Prince Valiant
© Respective copyright/trademark
holders.
|
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