HOPPY can help PreSchool Kids prepare to learn COMBINATORICS in PRIMARY, MIDDLE, and HIGH SCHOOL, by inducing Kids to teach HOPPY the INTORDUCTORY BASIC PRINCIPLES IN MATHTIVITIES. (Plant now. Reap or harvest later.")

In teaching HOPPY Combinatorics, Kidss teach HOPPY TO CHOOSE, which is the CRITICAL FIRST STEP OF DECISION-MAKING: LIFE ON-THE-FRONT-LINE. Now, COMBINATORICS is THE MATHEMATICS OF CHOICE -- how to CHOOSE REPRESENTATIVES IN A DEMOCRACY -- how to make WISE CONSUMER CHOICES IN A MARKET ECONOMY.

But, purely mathematically, COMBINATORICS has been called "counting without counting". This means that Kids teach HOPPY BASICS and some simple FORMULAS that IMMEDIATELY SUPPLY THE COUNT, AVOIDING A LONG TEDIOUS COUNTING PROCESS.

(Elsewhere, I note that ARITHMETIC WORKS BY SHORT-CUTS:

• COUNTING SHORTCUTS TALLYING;
• ADDING SHORTCUTS REPEATED COUNTING;
• MULTIPLYING SHORTCUTS REPEATED ADDITION;
• DIVISION SHORTCUTS REPEATED SUBTRACTION;
• EXPONENTIATION SHORTCUTS REPEATED MULTIPLICATION;
• LOGARITHM SHORTCUTS MULTIPLICATION, DIVISION, EXPONENTIATION, ROOT EXTRACTION.
• And COMBINATORIAL FORMULAS SHORTCUT THESE SHORTCUTS: SHORTCUTTING ON A SECOND LEVEL.)

BUT THE CONDITION OF INDEPENDENCE IS OFTEN NEEDED TO DO THIS AND THE MEANING OF CARTESIAN PRODUCT IS ALSO NEEDED.

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TEACHING HOPPY ABOUT INDEPENDENCE OF SETS OR EVENTS You can easily teah HOPPY to graph of "draw a picture" of INDEPENDENCE, one you see every day.

    *                              *
   /                               |     INDEPENDENT
  /     NONINDEPENDENT             |
 /_____                            |________

Imagne "dropping" a vertical segment from each asterisk. On the Left, the segment would touch the horizontal segment, as would every point of the segment the asterisk is in: DIFFERENT PROJECTIONS OF THE ASTERISK-SEGMENT ONTO THE OTHER SEGMENT; and these PROJECTIONS form A 1-D SUBSEGMENT OF HAT BOTTOM SEGMENT. However, on the RIGHT, the ONLY PROJECTION OF THE ASTERISK IS TO THE CORNER OF THE BOTTOM SEGMENT. But the corner is a POINT -- 1-D. Restated: GEOMETRIC NONINDEPENDENCE MEANS SHARING COMMON DIMENSION; GEOMETRIC INDEPENDENT MEANS SHARING ONLY DIFFERENT DIMENSIONS.

TWO SETS ARE INDEPENDENT IF, AND ONLY IF, THEY HAVE NO COMMON MEMBER [of the same kind].

TWO EVENTS, A, B ARE INDEPENDENT IF, AND ONLY IF, THE FOLLOWING FOUR CONDITIONS APPLY TO THEM:

1. A STATE EXISTS IN WHICH NEITHER EVENT OCCURS;
2. A STATE EXISTS IN WHICH A OCCURS, BUT B DOES NOT;
3. A STATE EXISTS IN WHICH B OCCURS, BUT A DOES NOT;
4. A STATE EXISTS IN WHICH BOTH A AND B OCCUR.

PLEASE NOTE THAT THESE ARE THE FOUR POSSIBLE STATES FOR TWO EVENTS. WHATEVER CAN HAPPEN, DOES HAPPEN -- INDEPENDENCE OF EVENTS.

In contrast, consider EVENTS C, D:

1. A STATE EXISTS IN WHICH NEITHER C NOR D OCCURS;
2. A STATE EXISTS IN WHICH C OCCURS BUT D DOES NOT;
3. NO STATE EXISTS WHEREIN D OCCURS BUT C DOES NOT;
4. A STATE EXISTS IN WHICH BOTH C, D OCCURS.

THEN EVENT D IS NOT INDEPENDENT OF EVENT C. IN FACT, EVENT D IS A SUBEVENT OF EVENT C, AS A VENN DIAGRAM SHOWS:

                  ---------------------------------------
                  |                C       *            |   *
                  |   -----------------------------     |
                  |   |            D       *       |    |
                  |   |                           |     |
                  |   |___________________________|     |
                  |_____________________________________|

NOTE: A POINT (ASTERISK) IN D IS ALSO IN C; BUT C HAS A PINT NOT IN D; AND THERE IS A POINT IN NEITHER. (THIS GRAPHS THE ABOVE CONDITIONS ON ONE KIND OF DEPENDENCE.)

COMBINATORISTS USE A TABLE TO EXPRESS THESE IDEAS, CALLED AN "INDICATOR TABLE". (A homologous Table in Statement Logic is called "a Truth-Table".) In an INDICATOR TABLE, entry "0" means "does not occur" (or "not in set"); entry "1" means "does occur" (or "in set").

Here are the INDICATOR TABLES FOR A, B -- and for C, D:

                A         B                        C          D
                0         0                        0          0
                0         1                        1          0
                1         0                        1          1
                1         1                       (2)        (1)
               (2)       (2)     

Elsewhere, I make use of a MEASURE I created, labelled "POLL": POLL MEASURES THE TOTAL COUNT OF ENTRIES IN AN INDICATOR COLUMN. Above, the numbers in parentheses represent POLL MEASURES.

Please note that events A and B both POLL 2, and their sum POLL is 2 + 2 = 4. (For math students, the POLL measures can be shown to obey the conditions for a METRIC, hence, SUMMING is MEANINGFUL.)

However, events C and D have different POLLS, with a SUM-POLL of only 3. Ome possibility is missing for C, D -- the one shown on Row 2 for the A-B Table.

My point: By COUNTING, and ADDING, HOPPY can DETERMINE INDEPENDENCE OF EVENTS OR SETS -- a critical COMBINATORIAL property.

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FLOWCHARTING INDEPENDENCE

                      _______
                      \START/
                       \   /
                        \ /
                         |
                         v
                       /  \
                 _____v_   v_____
                 |POLL A| |POLL B|
                 -------- --------
                      \    /
                       v  v
                        \/                                                            /\
                        / \                         _______________________          /  \
                       /ARE\--------NO----->--------|PRINT SETS INDEPENDENT|---->---/STOP\
                      /BOTH \                       ------------------------        ------
                      \ 2?  /                                                         ^
                       \   /                        _______________________           |
                        \ /---------YES---->--------|PRINT SETS INDEPENDENT|-----------
                         *                            ------------------------

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USING INDEPENDENCE TO TEACH HOPPY THE COMBINATORIAL SUM-RULE (ADDITION-RULE):

1. YOU CONSIDER TWO INDEPENDENT PROCESSES OR EVENTS;
2. THE FIRST PROCESS OR EVENT CAN HAPPEN IN n DIFFERENT WAYS;
3. THE SECOND PROCESS OR EVENT CAN HPPEN IN m DIFFERENT WAYS;
4. THEN THE TWO PROCEESES OR EVENTS, PUT TOGETHER, CAN (BY INDEOENDECE) HAPPEN IN n + m DIFFERENT WAYS.

How do we teach this to HOPPY? Well, elsewhere, we taught HOPPY how to COUNT out a NUMBER of ITEMS, say, n of them or n. And we also taught HOPPY how to ADD TWO NUMBERS, such as n + m. So, we just put HOPPY through these MATHTIVITIES in a way that teaches HOPPY THE COMBINATORIAL SUM-RULE.

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USING INDEPENDENCE TO TEACH CARTESIAN PRODUCT OF TWO SETS

Given set A = {dog, cat}, B = {flower, tree, bush}, their CARTESIAN PRODUCT -- denoted A X B -- is THE SET OF ALL ORDERED PAIRS FORMED BY FIRST MEMBER OF PAIR FROM FIRST SECOND, SECOND MEMBER OF PAIR FROM SECOND SET, IN ALL THE POSSIBLE WAYS.

S X T = {[dog, flower], [dog, tree], [dog, bush], [cat, flower], [cat, tree], [cat, bush]}. Please note that, for 2 members in set S, 3 members in set T, we have a total of 2 X 3 = 6 ORDERED PAIRS.

Elsewhere, we show how to teach HOPPY how to MULTIPLY by FORMING A TABLE WITH ROW EQUAL TO FIRST MULTIPLIER, WITH COLUMNS EQUAL TO SECOND MULTIPLIER, AND HOPPY COUNTS THE TOTAL FOR THE PRODUCT. So, this emulates the abstract pattern in the above Cartesian Product. Hence, HOPPY knows the MATH BEHIND THE COMBINATORIAL PRODUCT RULE.

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USING INDEPENDNCE AND CARTESIAN PRODUCT TO TEACH HOPPY THE COMBINATORIAL PRODUCT RULE:

1. Given INDEPENDENT EVENTS A, B;
2. A can occur in n ways (say 2 ways);
3. B can occur in m ways (say 3 ways);
4. then, by INDEPENDENCE, EVENTS A, B can occur in n X m ways (say 2 X 3 = 6 ways).

Given the problem of OBSERVING THE ABOVE SETS OF ANIMALS AND FLOWERS, HOPPY can use THE TABLE METHOD TO CALCULATE A COUNT OF 6, for such a case.

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We can teach HOPPY one of the main tools of COMBINATORICS: PERMUTATIONS. (PERMUTATIONS COMPOSE THE FINGER GROUP, shown at this Website.)