Fuzzy set 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.
Standard fuzzy set operations
Let A and B be fuzzy sets that A,B ⊆ U, u is any element (e.g. value) in the U universe: u ∈ U.
- Standard complement
The complement is sometimes denoted by ∁A or A∁ instead of ¬A.
- Standard intersection
- Standard union
so that for all x,y ∈ [0, 1] the following holds true:
- u(x,y) = n( i( n(x), n(y) ) )
(generalized De Morgan relation). This implies the axioms provided below in detail.
μA(x) is defined as the degree to which x belongs to A. Let ∁A denote a fuzzy complement of A of type c. Then μ∁A(x) is the degree to which x belongs to ∁A, and the degree to which x does not belong to A. (μA(x) is therefore the degree to which x does not belong to ∁A.) Let a complement ∁A be defined by a function
- c : [0,1] → [0,1]
- For all x ∈ U: μ∁A(x) = c(μA(x))
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]
c is a strong negator (aka fuzzy complement).
A function c satisfying axioms c1 and c2 has at least one fixpoint a* with c(a*) = a*, and if axiom c3 is fulfilled as well there is exactly one such fixpoint. For the standard negator c(x) = 1-x the unique fixpoint is a* = 0.5 .
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].
- For all x ∈ U: μA ∩ B(x) = i[μA(x), μB(x)].
Axioms for fuzzy intersection
- Axiom i1. Boundary condition
- i(a, 1) = a
- Axiom i2. Monotonicity
- b ≤ d implies i(a, b) ≤ i(a, d)
- Axiom i3. Commutativity
- i(a, b) = i(b, a)
- Axiom i4. Associativity
- i(a, i(b, d)) = i(i(a, b), d)
- Axiom i5. Continuity
- i is a continuous function
- Axiom i6. Subidempotency
- i(a, a) ≤ a
- Axiom i7. Strict monotonicity
- i (a1, b1) ≤ i (a2, b2) if a1 ≤ a2 and b1 ≤ b2
Axioms i1 up to i4 define a t-norm (aka fuzzy intersection). The standard t-norm min is the only idempotent t-norm (that is, i (a1, a1) = a for all a ∈ [0,1]).
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].
- For all x ∈ U: μA ∪ B(x) = u[μA(x), μB(x)].
Axioms for fuzzy union
- Axiom u1. Boundary condition
- u(a, 0) =u(0 ,a) = a
- Axiom u2. Monotonicity
- b ≤ d implies u(a, b) ≤ u(a, d)
- Axiom u3. Commutativity
- u(a, b) = u(b, a)
- Axiom u4. Associativity
- u(a, u(b, d)) = u(u(a, b), d)
- Axiom u5. Continuity
- u is a continuous function
- Axiom u6. Superidempotency
- u(a, a) ≥ a
- Axiom u7. Strict monotonicity
- a1 < a2 and b1 < b2 implies u(a1, b1) < u(a2, b2)
Axioms u1 up to u4 define a t-conorm (aka s-norm or fuzzy intersection). The standard t-conorm max is the only idempotent t-conorm (i. e. u (a1, a1) = a for all a ∈ [0,1]).
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]
Axioms for aggregation operations fuzzy sets
- Axiom h1. Boundary condition
- h(0, 0, ..., 0) = 0 and h(1, 1, ..., 1) = one
- 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.
- Klir, George J.; Bo Yuan (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall. ISBN 978-0131011717.
- Ismat Beg, Samina Ashraf: Similarity measures for fuzzy sets, at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016
- Günther Rudolph: Computational Intelligence (PPS), TU Dortmund, Algorithm Engineering LS11, Winter Term 2009/10. Note that this power point sheet may have some problems with special character rendering