# 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

- i is a t-norm,
- u is a t-conorm (aka s-norm),
- n is a strong negator,

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

*μ _{A}*(

*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*. (

*μ*(

_{A}*x*) is therefore the degree to which

*x*does not belong to

*∁A*.) Let a complement

*be defined by a function*

**∁**A*c*: [0,1] → [0,1]

- For all
*x*∈*U*:*μ*(_{∁A}*x*) =*c*(*μ*(_{A}*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
*x*∈*U*:*μ*_{A ∩ B}(*x*) =*i*[*μ*(_{A}*x*),*μ*(_{B}*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*(*a*_{1},*b*_{1}) ≤*i*(*a*_{2},*b*_{2}) if*a*_{1}≤*a*_{2}and*b*_{1}≤*b*_{2}

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* (*a*_{1}, *a*_{1}) = *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
*x*∈*U*:*μ*_{A ∪ B}(*x*) =*u*[*μ*(_{A}*x*),*μ*(_{B}*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* *a*_{1}<*a*_{2}and*b*_{1}<*b*_{2}implies*u*(*a*_{1},*b*_{1}) <*u*(*a*_{2},*b*_{2})

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 <
*a*_{1},*a*_{2}, ...,*a*_{n}> and <*b*_{1},*b*_{2}, ...,*b*_{n}> of*n*-tuples such that*a*_{i},*b*_{i}∈ [0,1] for all*i*∈*N*_{n}, if*a*_{i}≤*b*_{i}for all*i*∈*N*_{n}, then*h*(*a*_{1},*a*_{2}, ...,*a*_{n}) ≤*h*(*b*_{1},*b*_{2}, ...,*b*_{n}); that is,*h*is monotonic increasing in all its arguments.

- Axiom h3.
*Continuity* *h*is a continuous function.

## Further reading

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

## References

- 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