# Vagueness mathematics and fuzzy logic are employed, the

Vagueness is embroiled in
several real phenomena. Whether one contemplates vagueness explicitly when
modeling such a phenomenon is one of the modeling choices, the outcome of which
will rest on on the situation. On the other hand, the modeler chooses to reflect
vagueness, he or she will have to select the method for modeling it. Some
experts assert that one theory, e.g. probability theory, is adequate to model
every kind of vagueness. The excellence of the techniques used in a statistical
analysis rest on extremely on the presumed probability model or distribution.
Some physical systems,   those complex
ones, are uncompromising to model by an accurate and precise mathematical
procedure or equation due to the complexity of the system structure,
nonlinearity, uncertainty, randomness, etc.

Therefore, approximation
modeling is often necessary for real-world applications.  But, the important questions are what kind of
approximation is upright, where the logic of “goodness” has to be first
defined, of course, and how to formulate such a good approximation in modeling
a system such that it is mathematically rigorous and can produce satisfactory
results in both theory and applications.

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It is clear that interval
mathematics and fuzzy logic together can provide a promising alternative to
mathematical modeling for many physical systems that are too vague or too
complicated to be described by simple and crisp mathematical formulas or
equations. When interval mathematics and fuzzy logic are employed, the interval
of confidence and the fuzzy membership functions are used as approximation
measures, leading to the so-called fuzzy systems modeling.

Zimmermann H.-J. 134
proposed a definition of uncertainty: Uncertainty implies that in a certain
situation a person does not dispose about information which quantitatively and
qualitatively is appropriate to describe, prescribe or predict
deterministically and numerically a system, its behavior or other
characteristics. Buckley 12 define fuzzy probabilities, which will be fuzzy
numbers, from a set of confidence intervals and use fuzzy numbers for the
parameters in the probability density functions, to produce fuzzy probability
density functions.

Once dealing with the usual
probability concept, an occurrence has its specific limit. For take a
situation, if an event is X= {2, 4, 7, 9, 11}, its margin is sharp and
consequently, it can be characterized as a crisp set. As soon as we deal an
occurrence whose limit is not sharp, it can be measured as a fuzzy set, that
is, a fuzzy event. We can recognize the probability of two behaviors. One is
dealing with the probability of a crisp value (crisp probability) and the other
as a fuzzy set (fuzzy probability).

A hormone activity is a very
complex system, where it secretion processing is narrowly connected to
life-sustaining developments. It is evident now that there are a number of
amino acid combinations in the hormone that keep up its activity. Some details
of hormone activity, like stress or obesity, anxiety are well understood, while
others, like fetal growth, uterus contraction during cesarean and normal
delivery, primary postpartum hemorrhage effects and many others are still
ambiguous.

This means that a reasonable
explanation of the performance of a complex organism similar to a hormone’s
function is expressed in a natural-linguistic method with “sensitive” notations
like great excitability, weak damage, the low anticipation of punishment and so
on, which cannot be through statistically assigned.So, if we need to improve an
adequate tool for the logical theory of a hormone’s effective, in some sense,
we are bound to search for a mathematical tool, which could rightly work with
“perceptions” as with mathematical objects. Such mathematical tool was proposed
by Zadeh 131 and has been further developed during the last decades. It is
called fuzzy logic and fuzzy set theory. Guanrong Chen and Trung Tat Pham 50
give an example for fuzzy rule-based health monitoring expert systems, fuzzy
logic rules to control a focus ring of a camera to allow automatic focusing for
a sharp image, application of fuzzy control to a general class of servo
mechanic systems, the fuzzy controller for the robotic manipulator. The fuzzy
terms and definitions are defined in the following section 67, 135.

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