🦄 Fuzzy Mathematics

🐴 Fuzzy Logic & Sets

• 🐴 “Degrees of truth,” boundary-bending logic riddles.

Fuzzy mathematics is a form of alternative mathematics since it is based on a generalization of set membership

An element may belong 'to a degree' to a set

Vagueness

In different words, a vague concept is one that is characterized by fuzzy boundaries (i.e. there are cases where it is not clear if an object has or does not have a specific property or capacity

Sorites Paradox

Eubulides of Miletus

All agree that a single grain of wheat does not comprise a heap. The same applies to two wheat grains as they do not comprise a heap, etc. However, there is a point where the number of grains becomes large enough to be called a heap, but there is no general agreement as to where this occurs.

• The ontic view: In Nature
• The semantic view: In Language
• The epistemic view: In Sophisms, in knowing where the varier is.

∇𝜙: it is indeterminate whether 𝜙
Δ𝜙 : it is determined that 𝜙

Not the same as

• The term ambiguity refers to something that has more than one possible meaning, which may cause confusion
• Imprecision: When boundaries are not precise/defined.
• Uncertainty : unpredictable

one can approach borderline cases by using two nonvague solids: one that is contained in the vague object and the other that contains the vague object. These two nonvague objects are called the lower and the upper approximation, and they form a rough set

Fuzzy Sets

Aristotle's systematic study of logic

Gottfried Wilhelm Leibniz : Logic Algebra

Posets

Partially Ordered Set
Set P
relation ≤ 
- Reflexivity a ≤ a;
- Transitivity if a≤b and b≤c, then a≤c
- Antisymmetry if a≤b and b≤a, then a=b. 

One can say that each element of a poset is a logical proposition and the operator “≤” is the “⇒” operator. Thus, when x ≤ y, this actually means that x entails y.

greatest lower bound

(glb or inf or meet) for X if and only if:

• when x∈X, then y≤x, that is, y is a lower bound of X,and
•  when z is any other lower bound for X, then z ≤ y. Written
• y = ⋀ X

Least upper bound

(lub or sup or join) Same in the other side.

• y = ⋁ X.

Consider the set {0, 1}. This set is trivially a poset. The glb of this set is min(0, 1) = 0, while its lub is max(0, 1) = 1

Lattices

A poset P is a lattice if and only if every finite subset of P has both a glb and a lub.

In what follows, the symbol 𝟏 will denote the top (greatest) element and 𝟎 the bottom (least) element of a poset.

The operators ∧ and ∨ have the following properties:

• Commutativity
  • x ∧ y = y ∧ x
  • x ∨ y = y ∨ x
• Associativity
  • (x∧y) ∧ z = x ∧ (y∧z)
  • (x∨y) ∨ z = x ∨ (y∨z)
• Unit laws
  • x ∧ 𝟏 = x
  • x ∨ 𝟎 = x
• Idempotence
  • x ∧ x = x
  • x ∨ x = x
• Absorption
  • x ∧ (x ∨ y) = x
  • x ∨ (x ∧ y) = x

Frames

A poset P is a frame if and only if

1. every subset has a lub;

2. every finite subset has a glb;
3. the operator ∧ distributes over ∨: ⋁⋁ x ∧ ⋁Y = ⋁{x ∧ y|y ∈ Y}
 


Example: The set {false, true}, where false ≤ true, is a frame (why?).

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In classical set theory, one can define a set using one of the following methods:

• List method: A set is defined by naming all its members. This method can be used only for finite sets. For example, any set A, whose members are the elements a1 , a2 , ... , an , where n is small enough, is usually written as A = {a1,a2,...,an}.
Rule method. A set is defined by specifying a property satisfied by all its members. A common notation expressing this method is A = {x|P(x)}, where the symbol | denotes the phrase “such that,” and P(x) designates a proposition of the form “x has the property P.”





Characteristic function.

Assume that X is a set, which is called a universe, and A ⊆ X . Then, the characteristic function 𝜒A ∶ X → {0, 1} of A is defined as follows:

𝜒A(a)=
{  1, if   a ∈ A, 
{  0, if   a∉A. 

Zadeh had opted to introduce his fuzzy sets by employing an extension of the characteristic function:

Let X be a universe (i.e. an arbitrary set). A fuzzy subset A of X, is characterised by a function A ∶ X → [0, 1], which is called the membership function. For every x ∈ X, the value A(x) is called a degree to which element x belongs to the fuzzy subset A.

f there is at least one element y of a fuzzy subset A such that A(y) = 1, then A is called normal. Otherwise, it is called subnormal.

The height of a fuzzy subset A is the maximum membership degree, that is, h(A) = max{A(x)}, or h(A) = ⋁{A(x)} for infinite Universal Set X.

When the universe X is a set with few elements, it is customary to write down a fuzzy set A as follows:

A = a1∕x1 + a2∕x2 +···+ an∕xn,

where xi ∈ X, ai is the degree to which xi belongs to A, and ai∕xi means that xi belongs to A with a degree that is equal to ai. Obviously, the symbol “+” does not denote addition, but it is some sort of metasymbol. Alternatively, one can write down a fuzzy set as follows:

A = ∑ A(xi)∕xi n 
In case X is not finite (e.g. it is an interval of real numbers), a fuzzy set can be written down as

A= ∫ ai∕xi. X

Note that the symbols ∑ and ∫ do not denote summation or integration. In the first case, x ranges over a set of discrete values, while in the second case, it ranges over a continuum.

◼️	▢ 	▤	▩	◻️
 a	b	c	d	e

Assume that the set {a, b, c, d, e} forms a universe. Then, we can form a fuzzy subset of black squares of this set. Let us call this set B. Then, the membership values of this set are B(a) = 1, B(b) = 0.8, B(c) = 0.6, B(d) = 0.4, and B(e) = 0.3.

The empty fuzzy subset of some universe X is a set ∅ such that ∅(x) = 0, for all x∈X.

A fuzzy subset A of X such that A(x)=1 for all x∈X is called a crisp or sharp set (i.e. an ordinary set).

Assume that A ∶ X → [0,1] and B ∶ X → [0,1] are two fuzzy sets of X. Then,

• their union is 
(A ∪ B)(x) = max{A(x), B(x)}; 


• their intersection is 
(A ∩ B)(x) = min{A(x), B(x)};

• the complement of A is the fuzzy set 
A∁(x)=1−A(x), forall x∈X; 


• their algebraic product is 
(AB)(x) = A(x) ⋅ B(x); 


• A is a subset of B, denoted by A⊆B, if and only if 
A(x) ≤ B(x), for all x ∈ X; 


• the scalar cardinality of A is 
card(A) =∑ A(x) 


• fuzzy powerset of X 
𝓕(X). 


Corollaries:

• Commutation
• De morgan: ¢(A ∩ B) = ¢A ∪ ¢B

𝜶-Cuts of Fuzzy Sets

the 𝛼-cut [𝛼]A and the strong 𝛼-cut [𝛼+]A are the crisp sets [𝛼]A = {x|(x ∈ X) ∧ (A(x) ≥ 𝛼)} and [𝛼+]A = {x|(x ∈ X) ∧ (A(x) > 𝛼)}

An interesting property of 𝛼-cuts is that for any fuzzy set A ∶ X → [0, 1] if a1 < a2, then

[a1]A ⊇ [a2]A and [a1+]A ⊇ [a2+]A.

Theorem 2.3.1 Assume that A and B are two fuzzy subsets of X. T hen, for all a,b ∈ [0,1]

• [a+]A ⊆ [a]A;
• a = [a]A∩[a]B and a = [a]A∪[a]B;
• a+ = [a+]A∩[a+]B and a+ = [a+]A ∪ [a+]B
• a = ([(1−a)+]A)∁. 


Type 2 Fuzzy Sets: Intervaled

Just as meteorologists usually predict that the temperature will be within a specific range, it is more “natural” to use intervals as membership degrees. Thus, instead of using the set [0, 1], if we opt to use the set
 I([0, 1]) = {(a, b) | (a, b ∈ [0, 1]) ∧ (a ≤ b)}, where (a, b) is an open interval we can define an extension of fuzzy sets. Interval-valued fuzzy sets have been defined by Roland Sambuc:

An interval-valued fuzzy set A is a function A ∶ X → I([0, 1]), where X is some universe, and A(x) = [A,(x), A’(x)]. If (a,b),(c,d) ∈ I([0,1]), then (a,b) ≤ (c,d) if a ≤ c and b ≤ d.
 Assume that A, B ∶ X → I([0, 1]) are two interval-valued fuzzy sets. Then,

•  their union is 
(A ∪ B)(x) = [max(A,(x) , B,(x)) , max(A’(x) , B’(x))] ; 

•  their intersection is 
(A ∩ B)(x) = [min(A,(x) , B,(x)) , min(A’(x) , B’(x))] ; 

• the complement of A is the interval-valued fuzzy set 
A∁(x) = [1 − A’(x) , 1 − A.(x)] . 


“Intuitionistic” Fuzzy Sets and Their Extensions

An “intuitionistic” fuzzy set A is characterized by the expression {⟨x, 𝜇A(x), 𝜈A(x)⟩|x ∈ X}, where X is a universe, 𝜇A ∶ X → [0, 1] is a function called the membership function and 𝜈A ∶ X → [0, 1] is another function called the nonmembership function. Moreover, for all x ∈ X it must hold that 0 ≤ 𝜇A(x) + 𝜈A(x) ≤ 1.