Editing Fuzzy set (section)

===Disjoint fuzzy sets===
In contrast to the general ambiguity of intersection and union operations, there is clearness for disjoint fuzzy sets:
Two fuzzy sets <math>A, B</math> are '''disjoint''' iff
:<math>\forall x \in U: \mu_A(x) = 0 \lor \mu_B(x) = 0</math>
which is equivalent to
:[[Existential quantification#Negation|<math>\nexists</math>]] <math>x \in U: \mu_A(x) > 0 \land \mu_B(x) > 0</math>
and also equivalent to 
:<math>\forall x \in U: \min(\mu_A(x),\mu_B(x)) = 0</math>
We keep in mind that {{math|min}}/{{math|max}} is a t/s-norm pair, and any other will work here as well.

Fuzzy sets are disjoint if and only if their supports are [[disjoint sets|disjoint]] according to the standard definition for crisp sets.

For disjoint fuzzy sets <math>A, B</math> any intersection will give ∅, and any union will give the same result, which is denoted as
:<math>A \,\dot{\cup}\, B = A \cup B</math> 
with its membership function given by
:<math>\forall x \in U: \mu_{A \dot{\cup} B}(x) = \mu_A(x) + \mu_B(x)</math>
Note that only one of both summands is greater than zero.

For disjoint fuzzy sets <math>A, B</math> the following holds true:
:<math>\operatorname{Supp}(A \,\dot{\cup}\, B) = \operatorname{Supp}(A) \cup \operatorname{Supp}(B)</math>

This can be generalized to finite families of fuzzy sets as follows:
Given a family <math>A = (A_i)_{i \in I}</math> of fuzzy sets with index set ''I'' (e.g. ''I'' = {1,2,3,...,''n''}). This family is '''(pairwise) disjoint''' iff
:<math>\text{for all } x \in U \text{ there exists at most one } i \in I \text{ such that } \mu_{A_i}(x) > 0.</math>

A family of fuzzy sets <math>A = (A_i)_{i \in I}</math> is disjoint, iff the family of underlying supports <math>\operatorname{Supp} \circ A = (\operatorname{Supp}(A_i))_{i \in I}</math> is disjoint in the standard sense for families of crisp sets.

Independent of the t/s-norm pair, intersection of a disjoint family of fuzzy sets will give ∅ again, while the union has no ambiguity:
:<math>\dot{\bigcup\limits_{i \in I}}\, A_i = \bigcup_{i \in I} A_i</math> 
with its membership function given by
:<math>\forall x \in U: \mu_{\dot{\bigcup\limits_{i \in I}} A_i}(x) = \sum_{i \in I} \mu_{A_i}(x)</math>
Again only one of the summands is greater than zero.

For disjoint families of fuzzy sets <math>A = (A_i)_{i \in I}</math> the following holds true:
:<math>\operatorname{Supp}\left(\dot{\bigcup\limits_{i \in I}}\, A_i\right) = \bigcup\limits_{i \in I} \operatorname{Supp}(A_i)</math>