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### Fuzzy Sets

Week 2

Fuzzy sets is fully defined by its membership functions.

Membership function is a function in [0,1] that represents the degree of belonging.

Let’s start with a simple example

- Ask yourself how tall you are?
- Would you classify yourself as a tall person?
- What is the limit that determines tall and short people?
- Let’s collect some figures from you

“Because Fuzzy Logic is similar to the way we talk and think, it is easier for us to adjust” Cynthia Taylor

- Fuzzy Logic is particularly good at handling uncertainty, vagueness and imprecision.
- This is especially useful where a problem can be described linguistically (using words) or, as with neural networks, where there is data and you are looking for relationships or patterns within that data.
- Fuzzy Logic uses imprecision to provide robust, tractable solutions to problems.
- Fuzzy logic relies on the concept of a fuzzy set.
- Zadeh introduced the notion of fuzzy sets in 1965 in a seminal paper (Fuzzy Sets, Information and Control, vol.8, pp:338-353). He is a Professor at the University of Southern California and is still active in the field. The idea of fuzzy sets described in his seminal work lays the basis for Fuzzy Logic.

0

5ft 11ins

7ft

height

Let’s consider the first example(How tall/short we are?)1

0

0

5ft 11ins

7ft

height

A crisp way of modelling tallness

A crisp version of short

Let’s consider the first example(How tall/short we are?)

1

tall

very tall

quite tall

0

5ft 11ins

7ft

height

crisp definitions for tallness

0.40

Definition in a Fuzzy Set(How tall/short we are?)tall

short

1

0

5ft 11ins

7 ft

height

Membership functions that represent tallness and short

Some maths! Formal definitions of a fuzzy set

- For any fuzzy set, (let’s say) A, the function µArepresents the membership function for which µA(x) indicates the degree of membership that x, of the universal set X, belongs to set A and is, usually, expressed as a number between 0 and 1

µA(x) : X [0,1]

- Fuzzy sets can be either discrete or continuous

The notation for fuzzy sets: for the member, x, of a discrete set with membership µ, we use the notation µ/x . In other words, x is a member of the set to degree µ.

- Discrete sets are defined as:

A = µ1/x1+µ2/x2+…..+µn/xn

- or (in a more compact form)

x1 ,x2 , ….. xn : members of the set A

µ1, µ2,…..µn: x1 , x2 ….. xn ’s degree of membership.

A continuous fuzzy setA can be defined as:

µ

Example: Discrete and Continuous fuzzy sets to represent the set of numbers “close to 1”

numbers

Middle-aged

- Example: describing people as “young”, “middle-aged”, and “old”
- Fuzzy Logic allows modelling of linguistic terms using linguistic variables and linguistic values. The fuzzy sets “young”, “middle-aged”, and “old” are fully defined by their membership functions. The linguistic variable “Age” can then take linguistic values.

1

young

old

0

Age

Key Points for a fuzzy set:

- The members of a fuzzy set are members to some degree, known as a membership gradeor degree of membership
- A fuzzy set is fully determined by the membership function
- The membership grade is the degree of belonging to the fuzzy set. The larger the number (in [0,1]) the more the degree of belonging.
- The translation from x to µA(x) is known as fuzzification
- A fuzzy set is either continuous or discrete.
- Fuzzy sets are NOT probabilities
- Graphical representation of membership functions is very useful.

- In many physical systems, measurements are never precise Fuzzy numbers are one way of capturing this imprecision by having a fuzzy set representing a real number where the numbers in an interval near to the number are in the fuzzy set to some degree.

Imprecision and

Vagueness or Linguistic Uncertainty

1

µ

The fuzzy number ‘About 35’

x

0

35

Vagueness or Linguistic Uncertainty

- Another use of fuzzy sets is where words have been used to capture imprecise notions, loose concepts or perceptions.

1

µ

The fuzzy set ‘Risk’

0

Risk

Lecture Fuzzy Operations

LAB Introduction to Fuzzy Logic Toolbox of MATLAB and Fuzzy Membership Functions

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