ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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One way to give the structure of an ordered set
is by means of its so-called "incidence matrix".
Suppose that the order relation in question is
notated as "x =< y". Then the incidence matrix
is a square array of 0's and 1's whose entries
are indexed by row value x and column value y,
with an entry of "1" if x =< y, otherwise "0".
Returning to ZOL(p, q), the next Table presents the incidence matrix
of the implication ordering <f_i => f_j> on the 16 boolean functions
f(p, q), to be easier on the eyes showing only the positive entries.
Table. Incidence Matrix of the Implication Ordering <f_i => f_j>
o-----o------o----------o---------------------------------------------------------------o
| | | f_i | f_j |
| p | 1100 | o---o---o---o---o---o---o---o---o---o---o---o---o---o---o---o---o
| q | 1010 | | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |
o-----o------o----------o---o---o---o---o---o---o---o---o---o---o---o---o---o---o---o---o
| | | | |
| f_0 | 0000 | () | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
| | | | |
| f_1 | 0001 | (p)(q) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_2 | 0010 | (p) q | 1 1 1 1 1 1 1 1 |
| | | | |
| f_3 | 0011 | (p) | 1 1 1 1 |
| | | | |
| f_4 | 0100 | p (q) | 1 1 1 1 1 1 1 1 |
| | | | |
| f_5 | 0101 | (q) | 1 1 1 1 |
| | | | |
| f_6 | 0110 | (p, q) | 1 1 1 1 |
| | | | |
| f_7 | 0111 | (p q) | 1 1 |
| | | | |
| f_8 | 1000 | p q | 1 1 1 1 1 1 1 1 |
| | | | |
| f_9 | 1001 | ((p, q)) | 1 1 1 1 |
| | | | |
| f_A | 1010 | q | 1 1 1 1 |
| | | | |
| f_B | 1011 | (p (q)) | 1 1 |
| | | | |
| f_C | 1100 | p | 1 1 1 1 |
| | | | |
| f_D | 1101 | ((p) q) | 1 1 |
| | | | |
| f_E | 1110 | ((p)(q)) | 1 1 |
| | | | |
| f_F | 1111 | (()) | 1 |
| | | | |
o-----o------o----------o---------------------------------------------------------------o
I think that that is kind of pretty.
Jon Awbrey
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