Fuzzy Logic – Set Theory

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Fuzzy Logic – Set Idea

Fuzzy units might be thought-about as an extension and gross oversimplification of classical units. It may be greatest understood within the context of set membership. Mainly it permits partial membership which implies that it comprise components which have various levels of membership within the set. From this, we will perceive the distinction between classical set and fuzzy set. Classical set incorporates components that fulfill exact properties of membership whereas fuzzy set incorporates components that fulfill imprecise properties of membership.

Fuzzy and Classical Set

Mathematical Idea

A fuzzy set A˜A~ within the universe of knowledge UU might be outlined as a set of ordered pairs and it may be represented mathematically as −

A˜=yUA~=y∈U

Right here μA˜(y)μA~(y) = diploma of membership of yy in widetilde{A}, assumes values within the vary from zero to 1, i.e., μA˜(y)[0,1]μA~(y)∈[0,1].

Illustration of fuzzy set

Allow us to now take into account two circumstances of universe of knowledge and perceive how a fuzzy set might be represented.

Case 1

When universe of knowledge UU is discrete and finite −

A˜={μA˜(y1)y1+μA˜(y2)y2+μA˜(y3)y3+...}A~={μA~(y1)y1+μA~(y2)y2+μA~(y3)y3+…}

={ni=1μA˜(yi)yi}={∑i=1nμA~(yi)yi}

Case 2

When universe of knowledge UU is steady and infinite −

A˜={μA˜(y)y}A~={∫μA~(y)y}

Within the above illustration, the summation image represents the gathering of every factor.

Operations on Fuzzy Units

Having two fuzzy units A˜A~ and B˜B~, the universe of knowledge UU and a component of the universe, the next relations categorical the union, intersection and complement operation on fuzzy units.

Union/Fuzzy ‘OR’

Allow us to take into account the next illustration to know how the Union/Fuzzy ‘OR’ relation works −

μA˜B˜(y)=μA˜μB˜yUμA~∪B~(y)=μA~∨μB~∀y∈U

Right here ∨ represents the ‘max’ operation.

Union

Intersection/Fuzzy ‘AND’

Allow us to take into account the next illustration to know how the Intersection/Fuzzy ‘AND’ relation works −

μA˜B˜(y)=μA˜μB˜yUμA~∩B~(y)=μA~∧μB~∀y∈U

Right here ∧ represents the ‘min’ operation.

Intersection

Complement/Fuzzy ‘NOT’

Allow us to take into account the next illustration to know how the Complement/Fuzzy ‘NOT’ relation works −

μA˜=1μA˜(y)yUμA~=1−μA~(y)y∈U

Complement

Properties of Fuzzy Units

Allow us to focus on the completely different properties of fuzzy units.

Commutative Property

Having two fuzzy units A˜A~ and B˜B~, this property states −

A˜B˜=B˜A˜A~∪B~=B~∪A~
A˜B˜=B˜A˜A~∩B~=B~∩A~

Associative Property

Having three fuzzy units A˜A~B˜B~ and C˜C~, this property states −

(widetilde{A}cup left widetilde{B}) cup widetilde{C} proper = left widetilde{A} cup (widetilde{B}proper )cup widetilde{C})
(widetilde{A}cap left widetilde{B}) cap widetilde{C} proper = left  widetilde{A} cup (widetilde{B}proper cap widetilde{C})

Distributive Property

Having three fuzzy units A˜A~B˜B~ and C˜C~, this property states −

A˜(B˜C˜)=(A˜B˜)(A˜C˜)A~∪(B~∩C~)=(A~∪B~)∩(A~∪C~)
A˜(B˜C˜)=(A˜B˜)(A˜C˜)A~∩(B~∪C~)=(A~∩B~)∪(A~∩C~)

Idempotency Property

For any fuzzy set A˜A~, this property states −

A˜A˜=A˜A~∪A~=A~
A˜A˜=A˜A~∩A~=A~

Id Property

For fuzzy set A˜A~ and common set UU, this property states −

A˜φ=A˜A~∪φ=A~
A˜U=A˜A~∩U=A~
A˜φ=φA~∩φ=φ
A˜U=UA~∪U=U

Transitive Property

Having three fuzzy units A˜A~B˜B~ and C˜C~, this property states −

IfA˜B˜C˜,thenA˜C˜IfA~⊆B~⊆C~,thenA~⊆C~

Involution Property

For any fuzzy set A˜A~, this property states −

A˜¯¯¯¯¯¯¯¯=A˜A~¯¯=A~

De Morgan’s Legislation

This legislation performs a vital function in proving tautologies and contradiction. This legislation states −

A˜B˜¯¯¯¯¯¯¯¯¯¯¯¯¯=A˜¯¯¯¯B˜¯¯¯¯A~∩B~¯=A~¯∪B~¯
A˜B˜¯¯¯¯¯¯¯¯¯¯¯¯¯=A˜¯¯¯¯B˜¯¯¯¯