Fuzzy logic was initiated in 1965 by Lotfi A. Zadeh as a multivalued logic that allows intermediate values between evaluations like true/false. Fuzzy logic provides a more human-like way of thinking for computer programming. Unlike traditional binary logic, fuzzy systems use degrees of set membership between 0 and 1 rather than crisp 1 or 0 values. Key concepts include fuzzy sets which have membership degrees, and fuzzy operators like complement, union, and intersection that are defined based on membership degrees rather than binary outcomes. Fuzzy logic has been used to control complex systems and for applications like classification.

1. Fuzzy logic
 By: zaid da’ood

2. Fuzzy Logic Introduction
1.Introduction
Fuzzy Logic was initiated in 1965 [1], [2], [3], by
Lotfi A. Zadeh , professor for computer science at
the University of California in Berkeley.
Basically, Fuzzy Logic (FL) is a multivalued
logic, that allows intermediate values to be
defined between conventional evaluations like
true/false,yes/no, high/low, etc. Notions like rather
tall or very fast can be formulated
mathematically and processed by computers, in
order to apply a more human−like way of thinking
in the programming of computers[4].
Fuzzy systems is an alternative to traditional
notions of set membership

3. Fuzzy systems is an alternative to traditional notions of set
membership and logic that has its origins in ancient Greek
philosophy.
The precision of mathematics owes its success in large part
to the efforts of Aristotle and the philosophers who
preceded him.
In their efforts to devise a concise theory of logic, and
later a, the so−called "Laws of Thought" were posited
[5].

4. One of these, the "Law of the Excluded Middle," states that
every proposition must either be True or False. Even when
Parminedes proposed the first version of this law (around 400
B.C.) there were strong and immediate objections: for
example, Heraclitus proposed that things could be
simultaneously True and not True. It was Plato who laid the
foundation for what would become fuzzy logic, indicating that
there was a third region (beyond True and False) where these
opposites "tumbled about." Other, more modern philosophers
echoed his sentiments, notably Hegel, Marx, and Engels. But it
was Lukasiewicz who first proposed a systematic alternative to
the bi−valued logic of Aristotle [6].

5. Even in the present time some Greeks are still outstanding
examples for fussiness and fuzziness, (note the connection to
logic got lost somewhere during the last 2 mileniums [7]).
Fuzzy Logic has emerged as a a profitable tool for the
controlling and steering of systems and complex industrial
processes, as well as for household and entertainment
electronics, as well as for other expert systems and applications
like the classification of SAR data.

6. Fuzzy Sets
In contrast to crisp sets, fuzzy sets have membership
Degrees That means, in addition to the values 1 (belongs to)
and 0 (does not belong to) an element can have any value in
between (“kind of belongs to”) Formally speaking, the
membership function μA(x) for a fuzzy set A maps elements
x to any value in the interval [0, 1]:
μA : X → [0, 1]

7. Comparing Fuzzy Sets
 Equality:
 Crisp sets: two sets A and B are equal, if they
 contain the same elements
 Fuzzy sets: all membership degrees have to be
 equal, i.e. μA(x) = μB(x) for all x ∈ X
 Containment:
 Crisp sets: A is contained in B (A ⊆ B), if all
 elements in A are also elements of B
 Fuzzy sets: again membership degrees have to be
 considered, i.e. A ⊆ B ⇔ μA(x) ≤ μB(x) for all x ∈ X

8. Fuzzy Operators
 Complement (logical NOT): μA(u) = 1 − μA(u)
 Union (logical OR): μA[B(u) = max(μA(u), μB(u))
 Intersection(logical AND):μAB(u)=min(μA(u),μB(u))

9. Example: Build a Fuzzy Controller
FROM Fuzzy Thinkring, Bart Kosko
Goal: Design a motor speed controller for air
Conditioner.
Step 1: assign input and output variables
Let X be the temperature in Fahrenheit
Let Y be the motor speed of the air conditioner

10. Step 2: Pick fuzzy sets
Define linguistic terms of the linguistic
variables
temperature (X) and motor speed (Y) and
associate them with fuzzy sets
For example, 5 linguistic terms / fuzzy
sets on X
• Cold, Cool, Just Right, Warm, and Hot
Say 5 linguistic terms / fuzzy sets on Y
• Stop, Slow, Medium, Fast, and Blast

11. Input Fuzzy sets

12. Output Fuzzy sets

13. Step 3: Assign a motor speed set to
each
temperature set
• If temperature is cold then motor speed is stop
• If temperature is cool then motor speed is slow
• If temperature is just right then motor speed is
Medium
• If temperature is warm then motor speed is fast
• If temperature is hot then motor speed is blast

14. A Fuzzy Relation
expressed by a
rule

15. A Fuzzy controller with 5 patches

16. If temperature is just right then motor
speed is medium

17. t = 63 degree F.

18. Summary of Fuzzy Logic

19. • Rather than ealing with probability (0.0-1.0), Fuzzy
Logic deals with fuzzy set membership – an entity can
be in two or more sets at the same time, to different
degrees.
• Key concept is thus the degree of set membership
(0-1) The degree of set membership can also be referred
to as degree of truth of the proposition, X is in set Y.
• Degree of membership often calculated
algebraically, e.g., if height= 185, then degree = (185-170)
/(200-170) = 0.5

21. • Degree can also be given in tables (which would
give a stepped membership graph):

22. OPERATIONS:
• Subset: A fuzzy set A is a subset of fuzzy set B if for all
elements of A, the degree
of membership in A is less than in B.
• Complement: 2 fuzzy sets are complementary if for all
elements of A, the degree of
membership in B is 1-A(s).
• Union: The union of two fuzzy sets will assign degree of
membership equal to the
highest of the two sets.
• Intersection: The intersection of two fuzzy sets will
assign degree of membership
equal to the lowest of the two sets.

23. MEMBERSHIP IN COMPLEX SETS:
• To what degree is X in SET1 and SET2 ? – take
the lowest of the truth values (e.g.,as in
traditional logic A&B is false if any one is false).
• To what degree is X in SET1 or SET2 ? – take the
highest of the truth values (e.g., as in traditional
logic AvB is true if any one is true)

24. NATURAL LANGUAGE MODIFIERS

26. References
[1]L.A. Zadeh, Fuzzy Sets, Information and Control, 1965
[2]L.A. Zadeh, Outline of A New Approach to the Analysis of of Complex
Systems and Decision Processes, 1973
[3] L.A. Zadeh, "Fuzzy algorithms," Info. & Ctl., Vol. 12, 1968, pp. 94������ 102.
[4] L.A. Zadeh, "Making computers think like people," IEEE.
Spectrum, 8/1984,
pp. 26������ 32.
[5] S. Korner, "Laws of thought," Encyclopedia of Philosophy, Vol.
4, MacMillan,
NY: 1967, pp. 414������ 417.
[6] C. Lejewski, "Jan Lukasiewicz," Encyclopedia of Philosophy, Vol. 5,
MacMillan, NY: 1967, pp. 104������ 107.
[7] A. Reigber, "My life with Kostas", unpublished report, Neverending Story
Press , 1999

27. THANK YOU