🕡 🤦 ❇️ Fuzzy math. Basics of fuzzy sets ♎️ 🐙 💅

An excessive desire for accuracy began to exert an effect that negates control theory and system theory, since it leads to the fact that research in this area focuses on those and only those problems that can be precisely solved. Many classes of important problems in which data, goals, and constraints are too complex or poorly defined to allow accurate mathematical analysis have remained and remain on the sidelines only because they cannot be mathematically interpreted. L. Zade

Definition and characteristics

In the world, a lot is not divided only into white and black, into truth and truth ... A person uses many fuzzy concepts to evaluate and compare physical quantities, states of objects and systems at an approximate, qualitative level. So, any of us is able to estimate the temperature outside the window, without resorting to a thermometer, and guided only by our own feelings and a scale of approximate estimates (“overcast enough to take an umbrella”).

But a qualitative assessment does not possess the additivity property inherent in our usual numbers; that is, we cannot determine the result of operations for approximate estimates (“a small amount of money” + “a small amount of money”), in contrast to, for example, natural numbers (2 + 2). We can’t determine because a qualitative assessment strongly depends on the decision maker, context and meaning invested in a particular case.

However, in the world there are enough quantities that we are not able to accurately assess for one reason or another: the degree of order in the room, the "prestige" of the car, the beauty of a person, the "similarity" of things, ... But I want to work with them as usual numbers though would for automation tasks.

. 1964 .

( ) $\ tilde {A}$ () U $(u, \ mu_A (u))$ , $u \ subseteq U$ ,
$\ mu_A (u)$ — $\ tilde {A}$ , $\ mu_A (u): U → [0; 1]$ . $\ tilde {A} = \ bigcup _ {(u \ subseteq U)} (\ mu_A (u) / u) = \ left \ {(\ mu_A (u) / u) \ right \}$ .

U $\ mu_A (u)$ ( ) u (-) $\ tilde {A}$ . , . - .

$\ mu_A (u)$ ( . ), – . – U $\ tilde {A}$ . , , .

, $\ mu_A (u) = \ begin {cases} 1 & amp; u \ subseteq A \\ 0 & amp; u \ nsubseteq A \ end {cases}$ , .

, $\ mu_A (u)$ :

( );
;
;
;
...

-a≤x≤a.

“ ”.

$\ mu_A (x) = \ frac {(a- | x |)} {a}, -a \ leq x \ leq a$
$\tilde{A} = \left\{ 0 / -a;…;1 / 0;…;0.5 / \frac{a}{2};…;0 / a \right\}$ .

– , 1. 0.

, $\mu_A(u)$ 0.5, . -a/2 a/2.

$sup⁡(\mu_A(u)),u \subseteq U$ .

, 1, . – .

, , 0, .

, 1 .

2 $(-a, a)$ $\omega = \left\{x | \mu_A(x)>0, x \subseteq X \right\}$ – $\tilde{A}$ . $S_A$ $Supp A$ .

, x, $\mu_A(x)=0$ ; – $lim_{|x| \to \infty ⁡}{\mu_A (x)}=0$ .

, , . :

, ;
, ;
, , .

: $\tilde{A}, \tilde{B} ,\tilde{C}$ — U, $x \subseteq U$ . , .

$\tilde{A} = \tilde{B}$ , $\mu_A(x)= \mu_B(x)$ .

$A \subseteq B$ , $\mu_A(x) \leq \mu_B (x)$ x.

$\tilde{C} = \tilde{A} \cup \tilde{B}$ , $\mu_C (x) = max⁡ (\mu_A (x); \mu_B (x))$ . , $\mu_C(x) = \mu_A(x) \vee \mu_B (x)$ . (t– s–)

$\tilde{C} = \tilde{A} \cap \tilde{B}$ , $\mu_C (x) = min⁡ (\mu_A (x); \mu_B (x))$ . , $\mu_C(x) = \mu_A(x) \wedge \mu_B (x)$ . (t-)

. , , . , min max .

$\tilde{C} = \tilde{A} \backslash \tilde{B}$ , $\mu_C (x) = \mu_A(x) - \mu_{A \cap B}(x) = \mu_A(x) - min⁡ (\mu_A (x); \mu_B (x)) = max(0; \mu_A(x) - \mu_B(x))$ .

$U \backslash \tilde{A}$ $\overline{\tilde{A}}$ . , $\mu_{A}(x) = 1- \mu_{\overline{A}}(x)$ .

. , , , (, $A \cap \overline{A}= ∅$ ). :

α- . α- $A_{\alpha}$ , $\ mu_A (x) \ geq \ alpha$ .

. $\ tilde {A} = \ bigcup_ {a \ subseteq M} {\ alpha * A _ {\ alpha}}$ , M — .

$A ^ {\ beta}$ , $\ mu_ {A ^ {\ beta}} (x) = \ mu_A ^ {\ beta} (x)$ . :

β = 2 ( CON(A) ). , . , “ ” ;
β = 0.5 ( DIL(A) ). , . “ ”.

$\ mu_ {A * B} (x) = \ mu_A (x) * \ mu_B (x)$ .

$\ mu_ {A \ circledcirc B} (x) = (\ mu_A (x) + \ mu_B (x) - 1) \ vee 0$ .

$\ mu_ {A \ triangle B} (x) = \ begin {cases} \ mu_B (x) & amp; \ mu_A (x) = 1 \\\ mu_A (x) & amp; \ mu_B (x) = 1 \\ 0 & amp; \ end {cases}$ .

$\ mu_ {A + B} (x) = \ mu_A (x) + \ mu_B (x) - \ mu_A (x) * \ mu_B (x)$ .

$\ mu_ {A \ circledcirc B} (x) = (\ mu_A (x) + \ mu_B (x) - 1) \ wedge 1$ .

$\ mu_ {A \ triangledown B} (x) = \ begin {cases} \ mu_B (x) & amp; \ mu_A (x) = 0 \\\ mu_A (x) & amp; \ mu_B (x) = 0 \\ 1 & amp; \ end {cases}$ .

- – A B λ (1 — λ) ( A B). $\ mu_ {A _ + ^ {\ lambda} B} (x) = \ lambda * \ mu_A (x) + (1 - \ lambda) * \ mu_B (x)$ .

, λ- :

?

, , . . , ( , ). 2 – .

, . , , , . : , .

, , . , .

, , , , , :

0 <= μ(x) <= 1;
( );
The function and the set of defined functions must have a natural differentiation of concepts represented by neighboring sets;
There should not be gaps on a universal (or limited for consideration) set for which no set is associated;
For neighboring sets, the maximum of one must coincide with the minimum of the other, and the intersection point of their graphs must correspond to the transition points;
and some other task-specific ones.

although there are exceptional situations in which a function has to be determined based on context. The construction of such functions is a separate and rather complicated topic.

And that's all for today.

Fuzzy math. Basics of fuzzy sets

Definition and characteristics

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