To use a little more compact notation, let ZAND = the Zadeh AND, ZOR = the
Zadeh OR, LAND = the Lukasiewicz AND and LOR = the Lukasiewicz OR, with A LAND
B = max(A+B-1, 0) and A LOR B = min(A+B, 1).
You propose the following measures:
P = min(A(x), B(x)) = A ZAND B
Q = max(A(x), B(x)) = A ZOR B
You continue with
min(A \cap B) = min(P+Q, 1) = P LOR Q
max(A \cap B) = min(A, B) = A ZAND B
We can then state your final result as:
min(A \cap B) = P LAND Q = (A ZAND B) LAND (A ZOR B)
max(A \cap B) = A ZOR B
Using your examples:
A = .1
B = .2
P = A ZAND B = .1
Q = A ZOR B = .2
min(A \cap B) = .1 LAND .2 = 0
max(A \cap B) = .1 ZAND .3 = .1
A = .6
B = .8
P = A ZAND B = .6
Q = A ZOR B = .8
min(A \cap B) = .6 LAND .8 = .4
max(A \cap B) = .6 ZAND .8 = .6
yielding the same answers as you obtained.
However, the reasoning with which you achieved your operators is quite unclear
to me. Some clarification would be appreciated.
William Siler
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