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

In general, the triple (i,u,n) is called De Morgan Triplet iff

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 complements

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 xU: 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 .[2]

Fuzzy intersections

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 xU: μAB(x) = i[μA(x), μB(x)].

Axioms for fuzzy intersection

Axiom i1. Boundary condition
i(a, 1) = a
Axiom i2. Monotonicity
bd 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 a1a2 and b1b2

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 unions

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 xU: μAB(x) = u[μA(x), μB(x)].

Axioms for fuzzy union

Axiom u1. Boundary condition
u(a, 0) =u(0 ,a) = a
Axiom u2. Monotonicity
bd 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]).[2]

Aggregation operations

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 iNn, if aibi for all iNn, 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.

See also

Further reading

  • Klir, George J.; Bo Yuan (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall. ISBN 978-0131011717.


  1. Ismat Beg, Samina Ashraf: Similarity measures for fuzzy sets, at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016
  2. 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
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