Olin College of Engineering
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10-1-2010
Fall 2010 MTH 2110: Discrete Mathematics:
Course Materials: Class Notes: Fuzzy Sets and
Logic
Sarah Spence Adams
Olin College, [email protected]
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Recommended Citation
Adams, Sarah Spence, "Fall 2010 MTH 2110: Discrete Mathematics: Course Materials: Class Notes: Fuzzy Sets and Logic" (2010).
All Course Material - Olin Course Repository. Paper 102.
http://digitalcommons.olin.edu/course_repository/102
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Fuzzy Sets and Logic
Sarah Spence Adams
Discrete Mathematics
Rethinking Regular Sets

Universe U = {a, b, c, d, e, f, g}, set S = {a, c, g}

We can define S by mapping each element of
the universe to 0 or 1.

If we map it to 1, it goes in S. Else, not in S.

Here, f(a) =1, f(b) = 0, f(c)=1, f(d) = 0, f(e) = 0,
f(f) = 0, f(g) = 1.
Fuzzy Sets

Every element in a fuzzy set S has a
degree of membership

Map each element of U to a value
within the interval [0, 1]
S = { 0.6 a, 0.3 c, 0.9 g}
 c has 0.3 degree of membership in S

Fuzzy Logic

This business of having “degrees” of
membership rather than “in or out”

Truth values are between true and false

Introduced in 1965 to model uncertainty in
natural language: tall, fair, nice, large, hot
Why use fuzzy logic?
PROS:

Used to solve highly complex problems where math
modeling is too difficult/impossible

Tolerant of imprecise data

Approximation: can model arbitrary nonlinear functions

Intuitive, based on linguistic terms

Convenient way to express expert and common sense
knowledge
Why use fuzzy logic?
Cons:

Not a cure for all

Crisp/precise models can be more efficient
and even convenient

Other approaches might be formally verified
to work
Apply to Computer Games

Can have different characteristics of players


Strength: strong, medium, weak
Aggressiveness: meek, medium, nasty
• If meek and attacked, run away fast
• If medium and attacked, run away slowly
• If nasty and strong and attacked, attack back

Control of a vehicle



Should slow down when close to car in front
Should speed up when far behind car in front
Provides smoother transitions, no sharp boundary
Other Applications

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compensation against vibrations in camcorders
home appliances (washing machines, dish washers,
rice cookers, etc.)
recognition of handwriting, objects, voice
image processing
flight aid for helicopters
simulation for legal proceedings
improvement of fuel-consumption for automobiles
early recognition of earthquakes
“and in almost any other field you can think of”
Sound interesting?
It’s never to early to start thinking about
course project topics!