As an advantage over AXIOMATIC MATH, GENERATIVE MATH can
GRAPHICALLY GENERATE PATTERNS.
GENERATIVE MATH GIVES YOU CONTROL TO CONSTRUCT MATHEMATICS, thereby REALIZING its
PATTERN-STRATEGY-SELFPROSTHETIC essence.
The BASIC PATTERM-MAKER of DISCRETE MATHEMATICS is THE CARTESIAN PRODUCT.
The CARTESIAN PRODUCT constructs THE PATTERNS of ALL RELATIONS, FUNCTIONS, OPERATIONS.
The PATTERN constructed by THE CARTESIAN PRODUCT is familiar to you as THE PATTERN OF THE
CALENDAR MONTH:
- WEEKS as ROWS;
- WEEKDAYS as COLUMNS;
- DATES of MONTH as CELLS of THE PATTERN.
Given the SET {a,b,c,d,e},
the CARTESIAN PRODUCT of this SET with ITSELF is a SQUARE PATTERN:
{a,b,c,d,e} X {a,b,c,d,e}.
The RESULT of ANY CARTESIAN PRODUCT is a SET, such that EACH MEMBER is
AN ORDERED PAIR (a.k.a. 2-tuple, written with brackets: [1st,2nd]) wherein the PAIR'S 1ST MEMBER
is from the PRODUCT'S 1ST SET (on left-side of written form) and the
PAIR'S 2nd MEMBER is from the PRODUCT'S 2ND SET.
Thus, the CARTESIAN PRODUCT of the SET
{a,b,c,d,e} with ITSELF is: {a,b,c,d,e} X
{a,b,c,d,e} = {[a,a],[a,b],[a,c],...,[b,a];,[b,b],[b,c],...,[e,a],[e,b],...,[e,d],[e,e]}
, with 25 ORDERED PAIRS, from 5 original MEMBERS.
(ORDERED PAIRS THAT GENERATE A CARTESIAN PRODUCT ARE LIKE PIXELS THAT GENERATE NEWSPHOTOS OR
SCREEN GRAPHICS.)
The SQUARE CARTESIAN PRODUCT has important SUBPATTERNS, such as its DIAGONAL (WHITE PAIRS IN
TABLE BELOW). THE DIAGONAL FEATURES ELEMENTS IN 1st ROW & 1st COLUMN, 2nd ROW & 2ND COLUMN, ETC.
THAT IS, DIAGONAL PAIRS HAVE THE SAME 1ST & 2ND MEMBERS.
The advantage of GENERATIVE MATH pops out here.
- Given the SELF-CARTESIAN-PRODUCT, it GRAPHS as a SQUARE.
- The SUB-SELF-CARTESIAN-PRODUCT (1ST & 2ND MEMBERS the same) GRAPHS as the SUBPATTERN which
is THE DIAGONAL of that SQUARE.
- That SUBPATTERN EXPOSES TWO MORE SUBPATTERNS OF THE SQUARE:
- A TRIANGULAR SUBPATTERN (in blue font) ABOVE THE (whitened) DIAGONAL;
- A TRIANGULAR SUBPATTERN (in orange font) BELOW THE (whitened) DIAGONAL.
- EVERY BLUED PAIR ABOVE THE DIAGONAL -- SAY, [b,a] -- HAS AN ASSOCIATED ORANGED PAIR -- [a,b]
-- ABOVE THE DIAGONAL, as its INVERSE PAIR.
- THESE SUBGRAPHS, in turn, RECOGNIZABLY INDUCE SUBRELATIONS.
- THE DIAGONAL INDUCES and GRAPHS a REFLEXIVE RELATION< -- ONE THAT A
RELATUM HAS TO ITSELF. (Examples: In arithmetic, a number equals itself; in
geometry, a geometric figure is congruent to itself.)
- The UPPER and LOWER TRIANGLES INDUCE and GRAPH a SYMMETRIC
RELATION -- ONE THAT TWO DIFFERENT ELMENTS or PATTERNS MAY HAVE TO EACH OTHER. (Example in
arithmetic: 1 + 2 = 3; 3 = 1 + 2.)
Hey! REFLEXIVITY and SYMMETRY are 2 of the 3 CONDITIONS (a.k.a. PROPOERTIES)
for an EQUIVALENCE RELATION -- perhaps the most important RELATION IN
MATHEMATICS AND SCIENCE, since (as we'll soon see)
an EQUIVALENCE RELATION BUILDS BOTH THE MEASURES of TYPE&ORDER.
However, the EUIVALENCE RELATION ALSO REQUIRES another PROPERTY, namely, TRANSITIVITY:
If & are PRESENT (a.k.a., HOLD), THEN also is . ("If you can wald down Main
Street from Charlie Avenue to Snoopy Avenue, thence walk down Main Street from Snoopy Avenue to
Woodstock Avenue, then you can walk down Main Street from Charlie Avenue to Woodstock Avenue.
Dig?") But, hey! hey! We already have that present in a Self- or Square-Cartesian Procuct!
fOR, by DEFINITION, every element of the SET -- used to construct this Product --
(repeating) every element of it APPEARS BOTH AS 1ST & 2ND MEMBER OF THE PAIRS IN THE
PRODUCT. So the RELATION GRAPHED BY THAT SQUARE DIAGRAM BELOW IS ALSO TRANSITIVE.
Well, hey! hey! hey! The SQUARE-CARTESIAN-PRODUCT CREATES an EQUIVALENCE RELATION, and its GRAPH
is a GRAPH of one! (See how rich you discover that you are? Yeah, when GENERATIVE MATH is your
"accountant". It's subtly hidden in the AXIOMATIC FORMAT of this, but takes an expert to expose
it!)
Do you see it in this Table? Dig the meaning?
GRAPH OF EQUIVALENCE RELATION
[a,a] | [a,b] |
[a,c] | [a,d] |
[a,e] |
[b,a] | [b,b] |
[b,c] | [b,d] |
[b,e] |
[c,a] | [c,b] |
[c,c] |
[c,d] | [c,e] |
[d,a] | [d,b] |
[d,c] | [d,d] |
[d,e] |
[e,a] | [e,b] |
[e,c] | [e,d] |
[e,e] |
- SEE! SPOT SEE! HOW GENERATIVE MATH GRAPHS!
- SEE! SEE! THE CARTESIAN PRODUCT PATTERN!
- SEE! SEE! THE DIAGONAL PATTERN OF REFLEXIVITY, THE UPPER SUBTRIANGLE of SYMMETRY, AND THE
SQUARED PATTERN OF TRANSITIVITY!
- SEE! SEE! THE PATTERN OF EQUIVALENCE!
And EQUIVALENCE BUILDS THE MEASURE of TYPE (a.k.a. KIND), whether in MATHEMATICS, BOTANY,
BIOLOGY, PHYSICS, etc. Two methods prevail:
- DECLARING DEFINITIVE CHARACTERISTICS (a.k.a. TRAITS, a.k.a. INDICATORS) of a given TYPE --
not always easy to use this method.
- set forth a PROTOTYPE: "Type T consists of whatever looks like prototype P". Dig? This
directly uses EQUIVALENCE RELATION.
NOW LET'S TURN TO SIMPLE OR TOTAL ORDERING, BY SEEING that
- STRICKING OUT THE DIAGONAL
- AND STRIKING OUT THE INVERSE-CORRESPONDENT OF ANY (UPPER SUBTRIANGLE) PAIR
ACHIEVES TRANSFORMATION OF THIS TABLE INTO ANOTHER WHICH PATTERNS A TOTAL ORDERING RELATION<
-- SUCH AS < OR >
-- THAT IS, LESS-OR-GREATER THAN -- SUCH AS SUBSET OR SUPERSET -- ETC..