Before illustrating the mechanisms which make fuzzy logic machines work, it is
important to realize what fuzzy logic actually is. Fuzzy logic is a superset of conventional(Boolean) logic that has been
extended to handle the concept of partial truth- truth values between
"completely true" and "completely false". As its name suggests, it is the logic underlying
modes of reasoning which are approximate rather than exact. The importance of
fuzzy logic derives from the fact that most modes of human reasoning and especially
common sense reasoning are approximate in nature.
Fuzzy Sets
Fuzzy Set Theory was formalised by Professor Lofti Zadeh at the University of California in 1965. What
Zadeh proposed is very much a paradigm shift that first gained acceptance in the Far East and its
successful application has ensured its adoption around the world.
A paradigm is a set of rules and regulations which defines boundaries and tells us what to do to be
successful in solving problems within these boundaries. For example the use of transistors instead of
vacuum tubes is a paradigm shift - likewise the development of Fuzzy Set Theory from conventional
bivalent set theory is a paradigm shift.
Bivalent Set Theory can be somewhat limiting if we wish to describe a 'humanistic' problem
mathematically. For example, Fig 1 below illustrates bivalent sets to characterise the temperature of a
room.
The essential characteristics
of fuzzy logic as founded by Zader Lotfi are as follows.
The third statement hence, define Boolean logic as a subset of Fuzzy logic.
This natural phenomenon can be described more accurately by Fuzzy Set Theory. Fig.2 below shows how fuzzy sets quantifying the same information can describe this natural drift.
young(x) = { 1, if age(x) <= 20,
(30-age(x))/10, if 20 < age(x) <= 30,
0, if age(x) > 30 }
A graph of this looks like:
Given this definition, here are some example values:
Person Age degree of youth -------------------------------------- Johan 10 1.00 Edwin 21 0.90 Parthiban 25 0.50 Arosha 26 0.40 Chin Wei 28 0.20 Rajkumar 83 0.00So given this definition, we'd say that the degree of truth of the statement "Parthiban is YOUNG" is 0.50.
Note: Membership functions almost never have as simple a shape as age(x). They will at least tend to be triangles pointing up, and they can be much more complex than that. Furthermore, membership functions so far is discussed as if they always are based on a single criterion, but this isn't always the case, although it is the most common case. One could, for example, want to have the membership function for YOUNG depend on both a person's age and their height (Arosha's short for his age). This is perfectly legitimate, and occasionally used in practice. It's referred to as a two-dimensional membership function. It's also possible to have even more criteria, or to have the membership function depend on elements from two completely different universes of discourse.
Fuzzy Set Operations.
The Union operation in Fuzzy set theory is the equivalent
of the OR operation in Boolean algebra.
The Intersection operation in Fuzzy set theory is the
equivalent of the AND operation in Boolean algebra.
Union
and
respectively is defined
as the maximum of the two individual membership functions. This is called
the maximum criterion.
Intersection and respectively is defined as the
minimum of the two individual membership functions. This is
called the minimum criterion. 
Complementis defined as the negation
of the specified membership function. This is caleed the negation
criterion.
The Complement operation in Fuzzy set theory is the equivalent of the NOT operation in Boolean algebra.
The following rules which are common in classical set theory also apply to Fuzzy set theory.
De Morgans
law
,
Associativity
Commutativity
Distributivity
Universe of Discourse
Fuzzy Set
Support
Crossover point
Fuzzy Singleton
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