1.3 The Basic Connectives 5
Fig. 1.4 Fuzzy sets Cold, Cool, Just Right, Warm and Hot used in a room tem-
perature control example
Example 1.8. Let us consider the cool fuzzy set as in the previous example.
A
2
(x)=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
0 if, 40 ≤ x<50
x−50
10
if 50 ≤ x<60
1 −
x−60
10
if 60 ≤ x<70
0 if 70 ≤ x ≤ 100
.
Its core is (A
2
)
1
= {60}, the
1
2
-level set is (A
2
)
1
2
=[55, 65], the α-level set is
(A
2
)
α
=[50+10α, 70−10α], 0 <α≤ 1 and the support is supp A
2
=(50, 70).
Remark 1.9. If the universe of discourse is a finite set X = {x
1
,x
2
, ..., x
n
}
then a fuzzy set A : X → [0, 1] can be represented formally as
A =
A(x
1
)
x
1
+
A(x
2
)
x
2
+ ... +
A(x
n
)
x
n
.
Example 1.10. Let us consider the expression “good grade in Mathematics”.
This expression can be represented as a fuzzy set G : {A, B, C, D, F }→[0, 1],
G =
1
A
+
0.7
B
+
0.3
C
+
0
D
+
0
F
. The core of G is G
1
= {A}, the support is
supp G = {A, B, C} and the
1
2
− level set is G
1
2
= {A, B}.
1.3 The Basic Connectives
Let F(X) denote the collection of fuzzy sets on a given universe of dis-
course X.
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