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l
A
\
T
B
ð
x
;
y
Þ¼
T
ð
l
A
ð
x
;
y
Þ
;
l
B
ð
x
;
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ÞÞ
l
A
[
S
B
ð
x
;
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S
ð
l
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nition composition of fuzzy relations
Let
X
,
Y
,
Z
be crisp sets, A, B binary fuzzy relations and T t-norm. Then sup-T
composition of fuzzy relations A and B is fuzzy relation C
¼
A
T
B with the
membership function
De
l
C
ð
x
;
z
Þ¼
sup
y
2
Y
T
ð
l
A
ð
x
;
y
Þ
;
l
B
ð
x
;
y
ÞÞ
:
3 Fuzzy Inference and Generalized Modus Ponens
The fuzzy inference is a process which is applied to reasoning based on vague
concept. The inductive method modus tollens and the deductive method modus
ponens are the basic rules of inference in binary logic. In modus ponens we infer
validity of a propositional formula q from validity of implication p
⇒
q and validity
of premise of a propositional formula p.
3.1 Generalized Modus Ponens
In fuzzy reasoning we use a generalized modus ponens according to following
statement, where A, B, A
′
are fuzzy sets, X, Y linguistic variables. The scheme
consists of a rule or a premise (prerequisite), an observing and a conclusion
(consequence).
Rule
, B
′
if X is A, then Y is B
Observing
X is A
′
Conclusion
Y is B
′
The observing does not have to correspond to the premise in the rule. According
to
finding degree of comparison between premise X is A in the rule and current
observing X is A
′
it happens modi
cation conclusion Y is B in the rule and getting
value B
′
of variable Y.IfitisA
′
= A in observing, it have to be valid B
′
= B. In fact,
we operate more rules, input and output variables.
Example:
Rule
if the slope is moderate, the bike trail dif
culty is easy
Observing
slope is steeper
Conclusion
bike trail dif
culty is harder
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