In the file "Diagnostic Testing"", you see some simple VENN diagrams -- SIMPLE because NO DOMAIN INTERNAL TO ANOTHER DOMAIN HAS A SECTION OUTSIDE OF IT ALSO. But the GENERAL VENN DIAGRAMS FOR THREE DOMAINS WITHIN A UNIVERSE OF DISCUSSION DOES.

 _________________________________
|     UNIVERSE OF DISCUSSION      |
| ___________________             | 
| A                  |            | 
| |         _________|__________  | 
| |        B         |          | | 
| |   _____|_________|_________ | | 
| |   |    |         |        | | |
| |___|____|_________|        | | |
|     |    |                  | | |
|     |    |__________________|_| |
|     |                       |   |
|     |_______________________C   |
|_________________________________|

The CLASS or SET OF A INTERSECTS WITH THE CLASSES OR SETS OF B AND C.

PROBLEM: HOW CAN WE DISCUSS WHAT IS ATTRIBUTIVE ABOUT ONE OF THESE CLASSES OR SETS WITHOUT INVOLVING THE OTHERS? In particular, HOW CAN WE DEAL WITH COUNTS OF MEMBERS IN ONE OF THEM WITHOUT INVOLVING THE OTHERS?

(This is one of my STANDARD TASKS, shown elsewhere, modeling what I tested my college students for. But it should be introduced earlier -- say, in ELEMENTARY GRADES. Briefly, over a 50 year period, I've seen many reports by CORPORATIONS and GOVERNMENT AGENCIES WHICH COULD NOT BE CORRECT! WHY? BECAUSE EACH OF THE "ERRANT" REPORTS CONTRADICTED ITSELF -- SAYING IT CONTAINED MORE OR LESS THAN IT CONTAINED! (Many years ago, some one called attention to a report about the student body at a Florida colleges. But the totals for gender and college-level did not sum to the TOTAL claimed. So the submitter asked, "Oh, golly, are THEY here also?")

For example, let's put some numbers in the above diagram:

 _________________________________
|    UNIVERSE                     |
| ___________________             | A contains 2 + 8 + 3 + 5 = 15 members.
| A    2             |  11        | B cpmtains 8 + 7 + 3 + 6 = 24 members. 
| |         _________|__________  | C contains 5 + 3 + 6 + 9 = 23 members.
| |        B    8    |   7      | | The UNIVERSE contains 11 + 2 + 8 + 7 + 8 + 3 + 6 + 9 = 54
| |   _____|_________|_________ | | members.
| |   |  5 |    3    |        | | | However the totals for A, B, C are 15 + 24 + 23 = 62, more
| |___|____|_________|        | | | than in the UNIVERSE. Why some counts get summed in twice, 
|     |    |         6        | | | and the count of 3 (in all three, A, B, C) gets summed in 
|     |    |__________________|_| | three times! How can we avoid this???
|     |              9        |   |
|     |_______________________C   |
|_________________________________|

The SOLUTION IS A FORM OF BOOLE'S THEOREM, which is discussed elsewhere -- in connection with GOOGOL. It is also known as THE INCLUSION-EXCLUSION PRINCIPLE.

The INCLUSION-EXCLUSION PRINCIPLE has a loconek or "everyday" CONNECTION, going back to prehistoric times, as I note in my poem MOMMA GEOMETRY. And potterers use the same technique today: SLAP ON SOME CLAY; SCAPE SOME OFF; SLAP ON MORE; SCRAPE A LITTLE OFF; etc.

Let's write N(A) = 15, N(B) = 24, N(C) = 23, N(U) = 54, for the counts noted above. Also, we see from the VENN DIAGRAM that 8 is INCLUDED IN BOTH A & B, so we write N(A ^^ B) = 8, where the symbol "^" denotes "set INTERSECTION: what both of two sets". (Intecsection in sets corresponds to MEET in LATTICE THEORY, discussed in PECKNG ORDERS.) Similarly, we find that N(A ^C) = 5, N(B ^ C) = 6, and N(A ^ B ^ C) = 3 (as noted above).

But what we really wish to know is N(A V B V C), where "V" denotes "UNION of sets: what is ine one or the other or both of two sets". (It corresponds to JOIN in LATTICES, seen in PECKING ORDER.)

THE INCLUSION-EXCLUSION PRINCIPLE IMPLIES THE FOLLOWING FORMULA:
N(A v B v C) = N(A) + N(B) + N(C) - N(A ^ B) - N(A ^ C) - N(B ^ C) + N(A ^ B ^ C).

Dig? The first three terms after the equal-sign would enter twice a count in any two of the sets and enter three times a count in all three sets. The three subtractive terms with cancel out any count entered twice. BUT!!! in removing any count in all three sets, it REMOVES THAT COUNT THREE TIMES, so -- at that point of the formula -- it is NO LONGER COUNTED, hence, must be put back in for its ONE TIME APPEARANCE IN THE LAST TERM! Dig?

MORAL: WHENEVER POSSIBLE, TEACH A CONCEPT OR PROCEDURE IN A FORM THAT CONNECTS IT WITH ANOTHER CONCEPT OR PROCEDURE. WHY? FOR AT LEAST TWO REASONS:

1. IF THE CONNECTION EXISTS, IT IS PART OF THE HISTORY OF ACCUMULATING KNOWLEDGE -- "Just gimmie the facts, Ma'm."
2. In "buying the one", the student is "already part owner of the second".