Geoscience Reference
In-Depth Information
1
1
0.8
0.8
A
0.6
0.6
A
or
B
B
0.4
0.4
0.2
0.2
0
0
(a)
(b)
1
1
A
and
B
0.8
0.8
Not (
A
or
B
)
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(c)
(d)
FIGURE 12.7
Representations of cross sections through fuzzy sets, comparable to the crisp set representa-
tions in Figure 12.2. (a) Two trapezoidal fuzzy sets
A
and
B
, (b) the
union
of the two, (c) the
intersect
and
(d) the
negation
. (From Fisher, P.F., Fuzzy modelling, in:
GeoComputation
, Openshaw, S. and Abrahart, R.,
eds., Taylor & Francis Group, London, U.K., 2000, pp. 161-186.)
this, it can be seen that the fuzzy operation is a generalisation of the Boolean operations (compare
Figures 12.2a through c and 12.7a through d). On the other hand, Equation 12.6 (see also Figure
12.7c) defines the
intersect
of the two fuzzy sets by taking the minimum value of μ for every mea-
sured value (Figure 12.7c). It is also directly comparable to the Boolean
intersect
(Figure 12.2c).
Finally, the
negation
operation (Figure 12.7d) is also a generalisation of the special case of the
Boolean operation (Equation 12.7):
(12.5)
µ
∪
=
max(
µ
,
µ
)
(
AB
)
() ()
A
B
(12.6)
µ
∩
=
min(
µ
,
µ
)
(
AB
)
() ()
A
B
(12.7)
µ
′
=−
1
µ
()
A
()
A
Indeed, Leung (1988) gives three further pairs of fuzzy
union
and fuzzy
intersect
operators which
all specialise to the correct versions of Boolean set operators, as well as a number of other operators
which do not, and goes on to state that one of the advantages of fuzzy logic is that there is no unique
definition of the laws of set combination. This is well illustrated by the exhaustive list of operators
presented by Zimmermann (2010).
Equations 12.5 and 12.6 and other equations that share the property of specialising to the
Boolean equivalent are known as the hard
union
and hard
intersect
operators. There are also
soft versions of the fuzzy
union
and fuzzy
intersect
operators (Equations 12.8 and 12.9, respec-
tively), and these bear a remarkable, and intended, similarity to the probability of an object being
a member of the crisp sets (Equations 12.2 and 12.1, respectively). These have the advantage
