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).[1] This implies the axioms provided below in detail.
Fuzzy complementsedit
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 complementsedit
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]
A function c satisfying axioms c1 and c3 has at least one fixpoint a* with c(a*) = a*,
and if axiom c2 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 .[2]
Fuzzy intersectionsedit
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 intersectionedit
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 for all 0 < a < 1
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]).[2]
Fuzzy unionsedit
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 unionedit
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 for all 0 < a < 1
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 union). The standard t-conorm max is the only idempotent t-conorm (i. e. u (a1, a1) = a for all a ∈ [0,1]).[2]
Aggregation operationsedit
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 setsedit
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.
Klir, George J.; Bo Yuan (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall. ISBN 978-0131011717.
Referencesedit
^Ismat Beg, Samina Ashraf: Similarity measures for fuzzy sets, at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016
^ abcGünther 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
L.A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965