This document discusses operations on fuzzy sets including fuzzy complementation, intersection, union, and general aggregation operations. It defines each operation and provides examples. For each operation, it also lists common axioms that define properties like boundary conditions, monotonicity, commutativity, associativity, and continuity.

1. Operations on Fuzzy Sets
S.KARTHIKEYAN
15 MCA 14

2. Topics
•™Fuzzy Operations
• Standard Fuzzy Operations
• Fuzzy Complementation
•™Fuzzy Intersection
•™Fuzzy Union
•™General Aggregation Operations

3. What is Fuzzy operations?
• A fuzzy set operation is an operation on fuzzy sets.
• These operations are generalization of crisp set operations.
• There is more than one possible generalization.
• The most widely used operations are called standard fuzzy set
operations.
• There are three operations
fuzzy complements, fuzzy intersections, and fuzzy unions.

4. Standard Fuzzy Operations

5. Fuzzy Complement

6. Definition
It defined as the collection of all elements in the universe that do not reside in the
set A.
𝐀 ̅ = { 𝐱/ 𝐱 /∉ 𝐀, 𝐱 ∈ 𝐗}
The complement of a set is an opposite of this set.
Let a complement cA be defined by a function
c : [0,1] → [0,1]
𝝁¬𝐀(𝐱) = 𝟏 − 𝝁𝐀(𝐱)

8. Example
• , if we have the set of tall men, its complement is the set of NOT tall men.
If A is the fuzzy set, its complement ¬A can be found as follows:
• 𝝁¬𝐀(𝐱) = 𝟏 − 𝝁𝐀(𝐱)
• Example: given a fuzzy set of tall men
• Tall men (0/180, 0.25/182.5, 0.5/185, 0.75/187.5, 1/190) of the fuzzy
set
• NOT tall men will be:
• NOT tall men (1/180, 0.75/182.5, 0.5/185, 0.25/187.5, 0/190)

9. Axioms for fuzzy complements
• Axiom c1. Boundary condition
c(0) = 1 and c(1) = 0
• Axiom c2. Monotonicity
For all a, b ∈ [0, 1], if a < b, then c(a) > c(b)
• Axiom c3. Continuity
c is continuous function.
• Axiom c4. Involutions
c is an involution, which means that c(c(a)) = a for each a ∈ [0,1]

10. Fuzzy intersections

11. Definition
• The intersection of two fuzzy sets A and B is specified in general
by a binary operation on the unit interval, a function of the form
i:[0,1]×[0,1] → [0,1].
𝛍𝐀∩𝐁 (𝐱) = 𝐦𝐢𝐧 [ 𝛍𝐀(𝐱),𝛍𝐁(𝐱)] = 𝛍𝐀(𝐱)∩ 𝛍𝐁(𝐱) , where 𝐱 ∈ 𝐗

13. Example
• A fuzzy intersection is the lower membership in both sets of each element. The fuzzy
intersection of two fuzzy sets A and B on universe of discourse X:
Example:
Tall men (0/165, 0/175, 0.0/180, 0.25/182.5, 0.5/185, 1/190)
Average men (0/165, 1/175, 0.5/180, 0.25/182.5, 0.0/185, 0/190)
The intersection of these two sets is:
Tall men ∩ average men (0/165, 0/175, 0/180, 0.25/182.5, 0/185, 0/190)
Or
Tall men ∩ average men (0/180, 0.25/182.5,0/185)

14. Axioms for fuzzy intersection
• Axiom i1. Boundary condition
(a n 1) = a
• Axiom i2. Monotonicity
b ≤ d implies (a n b) ≤ (a n d)
• Axiom i3. Commutativity
(a n b) = (b n a)
• Axiom i4. Associativity
(a n (b n d)) = (a n b) n d)
• Axiom i5. Continuity
i is a continuous function
• Axiom i6. Subidempotency
(a n a) ≤ a

15. Fuzzy unions

16. Definition
• The union of two fuzzy sets A and B is specified in general by a binary
operation on the unit interval function of the form
u:[0,1]×[0,1] → [0,1]
𝛍𝐀∪𝐁 ( 𝐱) = 𝐦𝐚𝐱 [ 𝛍𝐀( 𝐱), 𝛍𝐁( 𝐱)] = 𝛍𝐀( 𝐱)∪ 𝛍𝐁( 𝐱) , where 𝐱 ∈ 𝐗

18. Example
The union is the reverse of the intersection. That is, the union is the largest
membership value of the element in either set.
The fuzzy operation for forming the union of two fuzzy sets A and B on universe X
Example:
Tall men (0/165, 0/175, 0.0/180, 0.25/182.5, 0.5/185, 1/190)
Average men (0/165, 1/175, 0.5/180, 0.25/182.5, 0.0/185, 0/190)
The union of these two sets is:
Tall men ∪ average men (0/165,1/175,0.5/180, 0.25/182.5, 0.5/185, 1/190)

19. Axioms for fuzzy union
• Axiom u1. Boundary condition : (a u 0) = (0 u a) = a
• Axiom u2. Monotonicity : b ≤ d implies (a u b) ≤ (a u d)
• Axiom u3. Commutativity : (a u b) = (b u a)
• Axiom u4. Associativity : (a u(b u d)) = ((a u b) u d)
• Axiom u5. Continuity : u is a continuous function
• Axiom u6. Superidempotency : (a u a) ≥ a
• Axiom u7. Strict monotonicity : a1 < a2 and b1 < b2 implies (a1 U b1) < (a2 U b2)

21. Aggregation operations

22. Definition
Aggregation operations on fuzzy sets are operations by which several fuzzy sets
are combined in a desirable way to produce a single fuzzy set.
Aggregation operation on n fuzzy set (2 ≤ n) is defined by a function
h:[0,1]n → [0,1]

23. Axioms for aggregation operations fuzzy sets
Axiom h1. Boundary condition
h(0, 0, ..., 0) = 0 and h(1, 1, ..., 1) = 1
Axiom h2. Monotonicity
For any pair <a1, a2, ..., an> and <b1, b2, ..., bn> of n-tuples such that ai, bi ∈
[0,1] for all i ∈ Nn, if ai ≤ bi for all i∈ Nn, then h(a1, a2, ...,an) ≤ h(b1, b2, ..., bn); that
is, h is monotonic increasing in all its arguments.
Axiom h3. Continuity
h is a continuous function.