In YOU Þ MEASURE, a TABLE describes 3 measures: NAMING; TYPING; ORDERING.

I created the NAMING measure as what, in set-theoretic language is known as an "injection" (an ASSIGNMENT). Briefly, this means that EACH NAME APPLIES TO ONLY ONE NAMEABLE ("one kid to a desk"), although SOME NAMEABLES MAY BE NAMELESS ("empty desks"). For present purposes, this description of the NAMING measure is sufficient.

The set-theoretic nature of the above three measures, namely, NAMING and TYPING can be GRAPHED by a TRIANGLE.

In their seminal book, The Meaning of Meaning (1923), C. K. Ogden and I. A. Richards displayed "The Meaning Triangle":

                                CONCEPT
                                   *
                                  / \
                                 /   \
                                /     \
                               /       \
                              /         \
                             /           \
                            /             \
                           /               \
                    SYMBOL*-----------------* REFERENT

I adapt this device to present "The Set Triangle". For there are three ways of presenting a set:

1. NOMINATION (NAMING), as in "S" to designate "a set"
2. EXTENSION, as in {a, b, c, d} -- LISTING ITS MEMBERS
3. INTENSION, as {x|P(x)="first four letter of the alphabet"} -- writing in general, {x|P(x)}. Read this as "the set of all x such that the proposition P(x) is true of x".<

                               INTENSION ({x: P(x)}
                                   *
                                  / \
                                 /   \
                                /     \
                               /       \
                              /         \
                             /           \
                            /     SET     \
                           /               \
             NOMINATION (S)*-----------------*EXTENSION ({-,-,-,...})

NOMINATION or NAMING is easily understood. (Here's the inside "math-joke" about this. If a mathematician uses "M" to NAME a SET, "he must have received his graduate mathematical education where 'menge' is the word for "set'". If the mathematician uses "E" to NAME a SET, "he must have received his graduate mathematical education in France, where 'ensemble' is the word for "set'", French for 'set'". But if the mathematician uses "S" to NAME a SET, he must have received his graduate mathematical education in the United States". However, it's alleged that many mathematicians who have received their graduate mathematical education will use 'M' or 'E' to NAME a SET to give the impression they received their graduate mathematical education in Germany or France!)

EXTENSION is an obvious way to designate a SET -- LISTING ITS MEMBERS. WARNING: possible only for finite sets of modest membership.

So INTENSION is the most general way to designate a set, providing a set-builder {x: P(x)} with a RULE (P(X)) for recognizing its members, x.

(PLEASE NOTE!!! SET-BUILDER CAN BE USED TO STATE ALGORITHMS a.k.a. PROGRAMS FOR SOLVING PROBLEMS OR EXPLICATING SETS.)

Do you realize that you are PROTO-EDUCATED in the CONCEPTS of EXTENSION & INTENSION? That is, YOU KNOW THEM BUT MAY NOT KNOW YOU KNOW THEM. But you may remember that you used what I shall show to be equivalents for "extension" and "extension" in your English classes over all those years? Of course, you did! I'll show you by relabeling the previous triangle:

                              CONNOTATION (INTENSION)
                                   *
                                  / \
                                 /   \
                                /     \
                               /       \
                              /         \
                             /           \
                            /             \
                           /               \
                      WORD*-----------------*DENOTATION (EXTENSION)

For "connotation" is the "intension" or "meaning" of a "word", and "denotation" is its "usage".

As shown in the (previous) SET TRIANGLE, the intensional form of any set -- written {x|P(x)} -- provides (remember?) a RULE, P(x), for recognizing any member, x, of a given set.

This implies that a SETBUILDER can (INTENSIONALLY) DEFINE A TYPE or TYPOLOGICAL MEASURE.

And -- HEY! -- this, in turn, implies that the extension associated with this intension -- written in SET-FORM as {-,-,-,...} -- LISTS SOME OR ALL THE MEMBERS OR CARRIERS OF THIS EXPLICATED TYPE.

(Sad, but most human critters don't know this, and some have given me much grief over this, accusing me of wasting their time.)

A better way to explicate typological measure is to CONSTRUCT ITS REPRESENTATIVE as a Cartesian product of two sets, although its primary use is to CONSTRUCT ITS REPRESENTATIVE as a binary relation. And the latter can be used to CONSTRUCT THE REPRESENTATIVE of an equivalence relation, which is described in the above cited TABLE as leading to the GROUP associated with this MEASURE.

A BONUS is that the Cartesian-construction results in ORDERED PAIRS or VECTORS, which is implicit in the construction of an ORDERING MEASURE.

(I pause to NOTE the ADVANTAGE OF MATHEMATICAL LANGUAGE OVER ORDINARY LANGUAGE. ONE STRUCTURE MAY SERVE MANY PURPOSES. Given a little trouble in learning the math, and you're RICHER THAN YOU REALIZED!)

I'll briefly explain this, and write out the Cartesian product. Then we proceed to the next file for the NEXT STEP IN THAT CHAIN, namely, RELATIONS, leading to the STEPS after that, the FUNCTION and OPERATION.

BIG BONUS: RELATIONS, FUNCTIONS, OPERATIONS COMPRENDS MUCH OF MATHEMATICS!

(And this supports my "6 DEGREES OF SEPARATION" thesis about MATHEMATICS APPRECIATION.)

See, Spot, see! The Cartesian Product.