Name[setNam, Set[el1, el2,...]]
...assigns the NamedSet, setNam, to the set containing el1, el2,....
This Set assignment means, for example, that setNam[el1] will return True.
The following 3 Set-operations, (on the NamedSet "setNam"),
setNam[test],
Set[setNam][test] &
Not[Set[setNam]][test]...
...all return the same result depending on whether test is in setNam or not.
The difference is that setNam[test] never effects the contents of setNam,
while Set[setNam][test] adds test to setNam, if it is not a member,
and Not[Set[setNam]][test] removes test if it is a member.
The Set abstraction is used
to develop
the `Logic` concepts
"unchanging", "not", "intersection",
"union", "complement
...using...
True[...set...], Not[...set...], And[...set...], Or[...set...], Xor[...set...]
respectively.
Also see the English definition for "set".
See also the "god" named Set or Seth.
The keyword "Set " is used here as an Expression that matches a collection of distinct elements.
A Name with the Set Attribute, is protected from assignment or removal.
Sets are often Named objects. A Set's Name is its NamedSet, which is assigned as follows:
(1) Name[setNam, Set[el1, el2,...]]
...declares setNam as a Set containing the elements "el1, el2,..." (which may be any sequence of Expressions).
Once a NamedSet has been assigned in (1), it becomes a BooleanTest. Specifically,
setNam[el]
...will return True if el is an element in setNam.
Once a Set is Named, (as in (1)), it becomes an object that can aquire or lose members through Set operations. Specifically, if test is a potential element in the Named Set, setNam, then...
(2) Set[setNam][test]
...returns True if test is in setNam or,
...returns False if test is not in setNam, and incorporates test into setNam.
In either case, after (2) is executed, test
will be a member in setNam.
(2) is an acquisitive Set
operation.
By contrast, the dispositive Set operation is formed with the
inverse of Set[setNam].
If, test is any Expression,
(3) Not[Set[setNam]][test]
...returns True if test is in setNam and removes test from setNam. Alternately, it...
...returns False if test is not in setNam.
In either case, after (3) is executed, test will not be a member in setNam.
"setNam[test]", "Set[setNam][test]" and "Not[Set[setNam]][test]" all return the same result depending on whether test is in setNam or not. The difference is that "setNam[test]" never effects the contents in setNam, while "Set[setNam][test]" adds test to setNam if it is not already a member, and "Not[Set[setNam]][test]" removes test if it is a member.
Both (2) and
(3)
generalize to sequences
of input expressions. Specifically, if test
is
a sequence in (2)
or
(3), then a sequence
of Set operations is executed
on
each element (as described above) in the sequence.
Specifically,
Not[Set[setNam]][test1,
test2, ...]
...will test each, test1, test2, ..., for membership in setNam, and remove found members from setNam. Similarly,
Set[setNam][test1,
test2, ...]
...will test
each, test1, test2, ..., for membership
and add them
to setNam if they are not already members.
A Boolean Sequence
of test membership is the evaluated result of both of the above set
operations.
For most purposes, a Pattern and a Set are treated identically in Grok32`.
The PatternSet abstraction exploits this functional similarity.
Each of the Logic` keywords, {True, Not, And, Or, Xor}, have PatternSet interpretations.
Specifically, the Set abstraction is used to develop the `Logic` concepts...
"unchanging", "not", "intersection", "union", "complement"
...using
True[...set...], Not[...set...], And[...set...], Or[...set...], Xor[...set...] respectively.
These elicitations and their functions are all specified in `Logic`Set.
If sym is any Name, then by default,
(4) Name[sym[Set]]
...is unassigned. Consequently, values may be assigned to sym without restriction. This will also be true, if sym has the Set Attribute assigned as follows:
If, on the other hand,
...then sym is protected, and any attempt to assign values to it will generate an error.
Only those definitions from Websters1949Unabridged
relating to the word "set" as a
keyword are included in the following.
set n.
A number of things of the same kind ordinarily used together; a collection of articles which naturally complement each other, and usually go together; an assortment; a suit; as, a set of chairs, or china, or books, of teeth, etc.
[From the 20th meaning of set (p. 2292 v2) in Websters1949Unabridged.]
set past tense & past part. of SET.
[Under division IV. A group or series set together...]
20. A number of things of the same kind ordinarily used together; a collection of articles which naturally complement each other, and usually go together; an assortment; a suit; as, a set of chairs, of china, of books, of teeth, etc.
21. A series associated by common authorship or publication. Specif., a collection of books forming a unit, as the works of one author issued in uniform style, a file of periodicals, related works on a particular subject, or unrelated books printed uniformly and intended to be sold as a group; as, a set of Dickens; a set of works on sociology.
22. A group of things naturally connected by location, formation, or order in time; as, a set of hills, muscles, teeth; a set of beats.
[23, 24, 25, 26, 27, 28, & 29 are all different types of sets formed by different set membership criteria...]
[The 40th definition is most precise...]
40. Math. The totality of all points or numbers that satisfy a given condition; the aggregate; as, the set of rational numbers.
[The only problem with the above definition is that it restricts itself to sets of points or numbers.
A set is the totality of objects that satisfy a given condition. The condition may be set membership, and other Boolean tests. JVWB]
[The Set Attribute, developed with the Name keyword, uses a different meaning for the word "set". The Set[...] elicitation refers to a noun. By contrast, the Set Attribute is best defined by the VIII category of "set" transitive verb definitions. Here is that definition:]
set v.t.
...
VIII. To put in a fixed state.
49. To fix firmly; to make fast, permanent, or stable; to render immobile; to give rigid form or condition to, as, to set one's jaw.
50. To make unyielding or obstinate; to render stiff, unpliant, or rigid; as, to set one's mind or heart.
51. To cause to stop or stick; as, to set a coach in the mud; hence, to puzzle to embarrass; perplex.
They show how hard they are set in this particular. Addision.
52. To render stiff or solid; esp., to convert into curd; to curdle; as, to set milk for cheese.
[The word "set" is ancient and incredibly rich. Just the definitions in Webster's paints the outline of an extraordinary history.]
Set (seËË�t), Seth (sa¯t), n.
Egypt Relig. An evil divinity, the brother and enemy of Osiris, represented with the head of a beast with high square ears and pointed snout. He was the oldest of all Egyptian gods, and when Horus came the gods divided Egypt, giving Lower Egypt to Horus and Upper Egypt to Set. Before the XIXth dynasty Set was popular, especially favored by the Hyksos, and probably regarded as a war god and giver of victories. About the XXXIId dynasty a reaction set in, the god's images were destroyed or defaces, and the became the personification of evil. The Greeks called him Typhon.
Typhon (ti¯?toËË�n) n. [L. , fr. Gr. Typho¯n. See Typhoon.]
Gr. & Rom. Myth. A monster, according to Hesiod the son of Typhoeus and the father by Echidna of Cerberus, the Chimera, the Sphinx, and other monsters. Later he is identified with Tyhoeus and by the Greeks with the Egyptian Set.
Typhoon (ti¯?fo¯o¯n?) n. [From Chin. (Cant.) tai-fung, (Pek.) ta-feng, liet., great wind, whence Jap. taifu¯, but influenced in E. by earlier tuphan, tufan, toofan, fr. Ar. tu¯fa¯n, fr. Gr. typho¯n, typho¯s,; akin to Gr. typhos smoke, vapor, Skr. dhu¯ma. See Fume; cf. TYHOID, TYHON.] A tropical cyclone occurring in the region of the Philippines or China Sea. See CYCLONE, n., 4.
[This is the end of the study of "Set"s theological significance as
defined in Websters1949Unabridged.
The ancient, international identification of "Set" and "Typhon"
suggests a myth-communicating, world-spanning culture dating back three
or four thousand years. Were these people the Polynesians or
which world
travelers? The fact that Set Theory has become, for many, the
foundation of Mathematics, and the fact that Physics
cannot be done without
this Mathematics, is evidence that the Set-cult
is very
much alive and well.
]
© 2004, 2005, 2006
by John Van Wie Bergamini
All rights reserved.