# Fuzzy set operations

Fuzzy set operations are a generalization of crisp set operations for fuzzy sets. There is in fact more than one possible generalization. The most widely used operations are called standard fuzzy set operations; they comprise: 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

    $\mu _{\lnot {A}}(u)=1-\mu _{A}(u)$
The complement is sometimes denoted by ∁A or A$^∁$ instead of ¬A.

### Standard intersection

    $\mu _{A\cap B}(u)=\min\{\mu _{A}(u),\mu _{B}(u)\}$

### Standard union

    $\mu _{A\cup B}(u)=\max\{\mu _{A}(u),\mu _{B}(u)\}$
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). 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 ∁A be defined by a function
    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 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 .

## 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 for all 0 < a < 1

#### 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]).

## 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 for all 0 < a < 1

#### 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 union). The standard t-conorm max is the only idempotent t-conorm (i. e. u (a1, a1) = a for all a ∈ [0,1]).

## 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.