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of fuzzy sets in the hypercube and there is no other paper on vector-valued variables
to characterize a property of a system.
An important step in the history of the fuzzy mathematics is Bart Kosko's work
on fuzzy sets as points in the hypercube. Kosko (Fig. 3.7 (a)) developed this theory
in the 1980s during his graduate studies in electrical engineering of the University
of California at Irvine and there he earned the Ph. D. in electrical engineering on
the foundations of fuzzy systems in 1987.
(a)
(b)
Fig. 3.7
(a): Bart Kosko in the 1990s.; (b) Headline of his article [35].
Therefore it took more than 15 years for this concept to became well-known
when Kosko established the “geometry of fuzzy sets” in a hypercube. Already from
the mid-eighties he wrote papers on his results and later he also published success-
ful topics on this subject and the article “Fuzziness vs. Probability” [35]. Kosko
intended to ostensive oppose this concept of the fuzzy hypercube to Zadeh's “sets-
as-functions definition of fuzzy sets” [35, p. 216]. He argued that the interpretation
of “fuzzy sets as membership functions, mappings
A
from domain
X
to range
[
,
]
”
is “hard to visualize. Membership functions are often pictured as two-dimensional
graphs, with the domain
X
misleadingly represented as one-dimensional.
In his 1990-paper “Fuzziness vs. Probability” (Fig. 3.7 (b)), Kosko wrote: “It
helps to see the geometry of fuzzy sets when discussing fuzziness. To date this
visual property has been overlooked. The emphasis has instead been on inter-
preting fuzzy sets as membership functions, mappings
A
from domain
X
to range
[
0
1
. But functions are hard to visualize. Membership functions are often pictured
as two-dimensional graphs, with the domain
X
misleadingly represented as one-
dimensional. The geometry of fuzzy sets involves both the
domainX
0
,
1
]
=
{
x
1
,...
x
n
}
and the
range
. The geometry of fuzzy sets is a
great aid in understanding fuzziness, defining fuzzy concepts, and proving fuzzy
theorems. Visualizing this geometry may by itself be the most powerful argument
for fuzziness.
The geometry of fuzzy sets is revealed by asking an odd question: What does the
fuzzy power set
F
[
0
,
1
]
of mappings
μ
A
:
X
→
[
0
,
1
]
2
X
(
)
, the set of all fuzzy subsets of
X
, look like? Answer: A cube.
