fuzzy logic - 爱码网

1. Classical (crisp) sets:
can be represented by a characteristic function(can only be zero or one):

for example:

In this situation, A is x>=4, so for the x like 0,1,2,3 is not belong to the set A, so the
2.Fuzzy sets:

can be represented by a membership function(can be [0, 1]):

Ps: (x-60/12) should be (x-60)/12
it would looks like:

3.Fuzzy sets or classical sets can be either discrete or continuous:

example: when x = x1, then = ( is also called the membership )

3.Fuzzy Sets and Probabilities

4. α-cuts
establishes a relationship between crisp sets and fuzzy sets.
For example:

={3,4,5,6} (because when x=3,4,5,6,>=0.2 )
(for the strong alpha cut , should > alpha)

The alpha cut(alpha=0.2) makes the fuzzy set A become a crisp set ,which =1 when x=3,4,5,6, otherwise zero

5. Support
The support of fuzzy set A is ={2，3，4，5，6，7}

6.Normality

The fuzzy sets is normalised when at least one element has height 1(if membership grades are in [0,1]).

For the fuzzy set A in (4), A is non-normal, as it doesn’t have one

6. Convexity
For a fuzzy set as below,

for any random r and s, any point between r and s, in this example, all the points in the yellow line, should be >= s (minimun of r and s)

since the yellow point which between the random r and s is smaller than the r (minimum of r and s), this fuzzy set is non-convex
For the fuzzy set A in (4), A is non-convex. when r is 0.7/3 and s is 0.7/5, the point 0.6/4 between r and s is smaller than r(or s), 0.6 < 0.7

·basic operations:

7.Complement

(not A)=(1-0)/1 + (1-0.1)/2 + (1-0.7)/3 + ...
Another example:

the fuzzy set ‘not young’ is the fuzzy complement of ‘young’

7. Intersection

=min(0, 0.1)/1 + min(0.1, 0.3)/2 + min(0.7, 0.6)/3 + ...

Another example:

’young’ ∩ ‘middle-aged’ = ‘young and middle-aged’

8. Union
=max(0, 0.1)/1 + max(0.1, 0.3)/2 + max(0.7, 0.6)/3 + ...
Another exmaple:

· parameterised operations:
Complement:

The function c must satisfy the following axioms
i.
ii. (非严格下降函数)

(a) and (b) are complement function, since they meet the axioms i and ii

what is the difference between parameterised operations and basic operations?
In my opinion, the basic operation just provide a basic function to convert a fuzzy set A to (by minusing by one, but actually, we can have other functions to converts A to as long as these functions satisfied the axioms above. )

9. Intersection

The intersection function can also be called as t-norms
The function must satisfy the following axioms

10.Union

The intersection function can also be called as t-conorms
The function must satisfy the following axioms

10.

12.

13.

14.

15. Mamdani Inference
Example:

We have three lingustic variables which are age, height, employ. For the age, it has three terms which are young, middle_aged, old. As shown in the left column, for example, the red line is the fuzzy set that used to represent the term young, the horizontal axis means the age of a person, the range of the vertical axis is between zero to one, if the value is one, then it means this person is young, and if the value is 0.5, then it means this person is half young and if the value is zero, then it means this person is not young. Lets take another example, the first picture of the right column. The horizontal axis means the salary a year, the range of the vertical axis is between zero to one, if the value is one, then it means this is a good job, and if the value is 0.5, then it means this is a half good job and if the value is zero, then it means this is not a good job.

Here is the fuzzy set:

Then, we have three rules:

IF Age is young AND Height is tall THEN he may get a good job

IF Age is middleaged THEN he may get a fair job

IF Age is old OR Height is short THEN he may get a bad job

And we give the inputs : age=40, height=1.8, we should output a number(how much salary will this person obtain?) or a term(the job is good, fair, or bad)

For the first rule:

we can get the output like this(the picture in the most right column):