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PART 4Fuzzy Arithmetic

FUZZY SETS AND FUZZY LOGIC Theory and

Applications

1. Fuzzy numbers 2. Linguistic variables 3.

Operations on intervals 4. Operations on fuzzy

numbers 5. Lattice of fuzzy numbers 6. Fuzzy

equations

Fuzzy numbers

- Three properties
- A must be a normal fuzzy set
- aA must be a closed interval for every
- the support of A, 0A, must be bounded.
- A is a fuzzy set on R.

Fuzzy numbers

Fuzzy numbers

- Theorem 4.1
- Let Then, A is a fuzzy number

if and only if there exists a closed interval - such that

Fuzzy numbers

- Theorem 4.1 (cont.)
- where is a function from

that is - monotonic increasing, continuous from the

right, - and such that

is a - function from that is

monotonic decreasing, continuous from the left,

and such - that

Fuzzy numbers

Fuzzy numbers

Fuzzy numbers

- Fuzzy cardinality
- Given a fuzzy set A defined on a finite

universal set X, its fuzzy cardinality, , is

a fuzzy number defined on N by the formula - for all

Linguistic variables

- The concept of a fuzzy number plays a fundamental

role in formulating quantitative fuzzy variables.

- The fuzzy numbers represent linguistic concepts,

such as very small, small, medium, and so on, as

interpreted in a particular context, the

resulting constructs are usually called

linguistic variables.

Linguistic variables

- base variable
- Each linguistic variable the states of which

are expressed by linguistic terms interpreted as

specific fuzzy numbers is defined in terms of a

base variable, the values of which are real

numbers within a specific range. - A base variable is a variable in the classical

sense, exemplified by any physical variable

(e.g., temperature, etc.) as well as any other

numerical variable, (e.g., age, probability,

etc.).

Linguistic variables

- Each linguistic variable is fully characterized

by a quintuple (v, T, X, g, m). - v the name of the variable.
- T the set of linguistic terms of v that refer

to a base variable whose values range over a

universal set X. - g a syntactic rule (a grammar) for generating

linguistic terms. - m a semantic rule that assigns to each

linguistic term t T.

Linguistic variables

Operations on intervals

- Let denote any of the four arithmetic

operations on closed intervals addition ,

subtraction , multiplication , and division /.

Then,

Operations on intervals

- Properties
- Let

Operations on intervals

Operations on fuzzy numbers

- First method
- Let A and B denote fuzzy numbers. denote any

of the four basic arithmetic operations. - for any
- Since is a closed interval

for each - and A, B are fuzzy numbers,

is - also a fuzzy number.

Operations on fuzzy numbers

- Second method

Operations on fuzzy numbers

Operations on fuzzy numbers

Operations on fuzzy numbers

- Theorem 4.2
- Let , -, , / , and let A, B denote

continuous fuzzy numbers. Then, the fuzzy set - AB defined by
- is a continuous fuzzy number.

Lattice of fuzzy numbers

- MIN and MAX

Lattice of fuzzy numbers

Lattice of fuzzy numbers

Lattice of fuzzy numbers

- Theorem 4.3
- Let MIN and MAX be binary operations on R.
- Then, for any , the

following properties hold

Lattice of fuzzy numbers

Lattice of fuzzy numbers

- Lattice
- It also can be expressed as the pair

, where is a partial ordering defined

as

Lattice of fuzzy numbers

Fuzzy equations

- A X B
- The difficulty of solving this fuzzy equation

is caused by the fact that X B-A is not the

solution. - Let A a1, a2 and B b1, b2 be two

closed intervals, which may be viewed as special

fuzzy numbers. B-A b1- a2 , b2 -a1, then

Fuzzy equations

- Let X x1, x2.

Fuzzy equations

- Let aA aa1, aa2, aB ab1, ab2, and
- aX ax1, ax2 for any .

Fuzzy equations

- A.X B
- A, B are fuzzy numbers on R. Its easy to

show that X B / A is not a solution of the

equation.

Exercise 4

- 4.1
- 4.2
- 4.5
- 4.6
- 4.9

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