1. Definition of fuzzy relationship
1. Direct product of sets
Given two sets X, Y, the direct product X×Y (Cartesian product) of X and Y is defined as:
It is a set on a two-dimensional domain of discourse composed of all the ordinal pairs (x, y). Generally X×Y≠Y×X.
2. Fuzzy relationship and fuzzy matrix
Assuming that X and Y are two non-empty sets, the fuzzy set R defined by the direct product X×Y is called the fuzzy relationship between X and Y, denoted as Rx×y.
(1) The fuzzy relation Rx×y is completely described by its membership function μR(x,y), and μR(x,y) indicates the degree to which the element x in X and the element y in Y have the relationship Rx×y.
(2) When X and Y are finite discrete sets, let X={x1, x2,...,xn}, Y={y1, y2,...,ym}, then the fuzzy relationship Rx×y of X and Y can be used n ×m order matrix (fuzzy matrix) representation, namely
A fuzzy matrix is a fuzzy set whose universe is the direct product X×Y.
Examples of fuzzy relations and fuzzy matrices are as follows
Example: X={10, 20, 40, 80}, Y={10, 20, 30, 40}, the fuzzy relationship matrix of the fuzzy relationship "x is much greater than y" is
Conclusion: When x=40, y=20, the degree of "x is much greater than y" is 0.8.
3. Direct product of fuzzy sets
If there are two fuzzy sets A and B, their domains are X and Y respectively, the fuzzy set A×B defined on the product space X×Y is called the direct product of fuzzy sets A and B, and its membership function is:
The direct product of fuzzy sets A and B is a fuzzy relation on the product space X×Y.
2. Synthetic operation of fuzzy relation and fuzzy matrix
Since fuzzy relations and fuzzy matrices are fuzzy sets defined in the direct product space, they obey the operation rules of general fuzzy sets (union, intersection, complement, etc.). (See 1 for details)
1. Composite operation of fuzzy matrix
Example introduction : known fuzzy relationship matrix
In summary, the row of R1 is compared with the column of R2, and the smallest one (from the equal sign in the second row to the equal sign in the third row) is found first, and then the largest number among the smallest is found as the final Results (from the third row to the fourth row).
2. The relationship between composite operation and direct product operation
Let A be a fuzzy set on domain X, and B be a fuzzy set on domain Y. According to the above definition of the direct product of fuzzy sets and the synthesis of fuzzy matrices, when X and Y are discrete domains, the direct product of A and B (take the small operation) is:
3. Composite Operation of Fuzzy Relation
Suppose R1 is the fuzzy relationship between X and Y, R2 is the fuzzy relationship between Y and Z, the combination of R1 and R2 R1○R2 refers to a fuzzy relationship on X×Z, and its membership function is:
In the above formula, "∨" represents a large operation, and "∧" represents a small operation, so it is called max-min composition.
PS: The metaphor of a small image, for a fuzzy relationship, its corresponding membership function is equivalent to its ID card.
When the universe of discourse X, Y, Z is a finite set, the composition of fuzzy relations can be represented by the composition of fuzzy matrices .
3. Fuzzy Logic and Fuzzy Reasoning
1. Fuzzy Linguistic Variables
Fuzzy linguistic variables are words or sentences in natural language, such as temperature, error, etc. Its value is not a usual constant, but a fuzzy set expressed in fuzzy language. Hereinafter, fuzzy linguistic variables are referred to as linguistic variables.
A linguistic variable can be represented by the following pentads:
x is the name of the language variable; T(x) is the set of language variable values; U is the domain of x; G is the grammatical rule (used to generate the name of each language variable value x); M is the semantic rule (used to generate fuzzy set membership function).
2. Fuzzy propositions
(1) Fuzzy proposition (proposition): a declarative sentence containing fuzzy concepts. Fuzzy propositions can be represented by English letters, such as P: the error is large.
(2) The truth value of the fuzzy proposition: the true or false degree of the fuzzy proposition, which is a real number on the interval [0,1] .
(3) Single fuzzy proposition: a simple fuzzy statement, whose general form is P: x is A where x is a fuzzy variable, and A is a fuzzy set corresponding to a certain fuzzy concept.
(4) The truth value V(P) of the single fuzzy proposition P is represented by the membership degree of the variable to the fuzzy set, namely
V(P)=μA(x)
When μA(x)=0, it means that the proposition P is completely false; μA(x)=1, it means that the proposition P is completely true; the closer μA(x) is to 0, the greater the degree of falseness of the proposition P, and the smaller the degree of truth; μA The closer (x) is to 1, the less false the proposition P is, and the greater the truth.
Example: Discuss fuzzy proposition Q: the weather is hot. The language variable is air temperature t, t belongs to [-40ºC, 50ºC], the fuzzy set defining "heat" is H, and its membership function is μH(t). If today's temperature is t=20 ºC, μH(20)=0.4, then the truth value of this proposition is 0.4. In other words, the true degree of the proposition "the weather is hot" is 0.4.
(5) Compound fuzzy propositions: Combine simple fuzzy propositions through connectives to form compound fuzzy propositions. The linking words can be "and", "or", "not", "if...then..." and so on.
(6) Conditional fuzzy proposition: A conditional statement in the form of "IF ... THEN ...", which expresses the causal relationship between two ordinary propositions, is called a conditional fuzzy proposition.
3. Fuzzy reasoning
Fuzzy inference is a kind of uncertainty reasoning method. It uses fuzzy language to make fuzzy judgments on fuzzy propositions and draw an approximate fuzzy conclusion.
(1) The basic form of fuzzy reasoning: syllogism
(2) Composition rules of fuzzy reasoning
In 1975, Zadeh proposed the composition rules of fuzzy logic reasoning.
That is, the conclusion B' is the synthesis of the fuzzy set A' and the fuzzy implication relation A→B.
Examples of Synthetic Inference Rules
If the furnace temperature is adjusted manually, there is a rule of thumb as follows: "If the furnace temperature is low, apply high voltage". What kind of voltage should be applied when the furnace temperature is "very low". It is known that x and y represent the fuzzy language variables "furnace temperature" and "voltage" respectively, and the domains of x and y are:
(3) Multiple fuzzy conditional statements
1) Fuzzy conditional statements connected with "and"
Fuzzy implication:
Inference result/operation method:
The steps are as follows: Find the fuzzy implication relation of the rule, find AXB; then find R;
Calculate the fuzzy set of the input quantity, find A', B';
Calculate the fuzzy set of the output quantity, find C'.
2) Fuzzy conditional statements using "also" joins
Inference result/operation method:
The steps are as follows: Find the fuzzy implication relation of each rule, find R1, R2;
Find the total fuzzy implication relation, find R;
Calculate the fuzzy set of the input quantity, find A'XB';
Calculate the fuzzy set of the output quantity, find C'.
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