1. Theory and Applications
FUZZY SETS AND
FUZZY LOGIC
Theory and Applications
PART 4 Fuzzy Arithmetic
1. Fuzzy numbers
2. Linguistic variables
3. Operations on intervals
4. Operations on fuzzy numbers
5. Lattice of fuzzy numbers
6. Fuzzy equations

2. Fuzzy numbers Three properties A is a fuzzy set on R.
A must be a normal fuzzy set;
αA must be a closed interval for every
the support of A, 0+A, must be bounded.
A is a fuzzy set on R.

3. Fuzzy numbers

4. Fuzzy numbers
Theorem 4.1
Let Then, A is a fuzzy number if and only if there exists a closed interval
such that

5. 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

6. Fuzzy numbers

7. Fuzzy numbers

8. 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

9. 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.

10. 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.).

11. 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.

12. Linguistic variables

13. Operations on intervals
Let ＊ denote any of the four arithmetic operations on closed intervals: addition ＋, subtraction —, multiplication • , and division /. Then,

14. Operations on intervals
Properties
Let

15. Operations on intervals

16. 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.

17. Operations on fuzzy numbers
Second method

18. Operations on fuzzy numbers

19. Operations on fuzzy numbers

20. Operations on fuzzy numbers
Theorem 4.2
Let ＊ {＋, －, •, / }, and let A, B denote continuous fuzzy numbers. Then, the fuzzy set
A＊B defined by
is a continuous fuzzy number.

21. Lattice of fuzzy numbers
MIN and MAX

22. Lattice of fuzzy numbers

23. Lattice of fuzzy numbers

24. Lattice of fuzzy numbers
Theorem 4.3
Let MIN and MAX be binary operations on R.
Then, for any , the following properties hold:

25. Lattice of fuzzy numbers

26. Lattice of fuzzy numbers
It also can be expressed as the pair , where is a partial ordering defined as:

27. Lattice of fuzzy numbers

28. 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

29. Fuzzy equations
Let X = [x1, x2].

30. Fuzzy equations Let αA = [αa1, αa2], αB = [αb1, αb2], and
αX = [αx1, αx2] for any

31. Fuzzy equations
A．X = B
A, B are fuzzy numbers on R+. It’s easy to show that X = B / A is not a solution of the equation.

32. Exercise 4
4.1
4.2
4.5
4.6
4.9