Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 126

Let us note that these concepts lead to a binary result. It is also possible to define
a degree of inclusion between two fuzzy subsets, but we will not discuss this here.

The intersection (the union, respectively) of two fuzzy subsets is defined by the
point-to-point minimum (maximum, respectively) between the membership functions:

ν )( x )=min μ ( x ) ,ν ( x ) ,

∀ ∈S ∩

x , ( μ [8.7]

ν )( x )=max μ ( x ) ,ν ( x ) . ∀ x ∈S , ( μ ∪ [8.8]

The complement of a fuzzy set is defined by:

,μ C ( x )=1 ∀ x ∈S − μ ( x ) . [8.9]

Here are the major properties of these operations:
- they are all consistent with set operations: in the specific case where the mem-
bership functions only have 0 and 1 as values (sets are then binary), these definitions
amount to the traditional binary definitions (this is an important property and the least
we would expect from the fuzzy extension of a binary operation);
- μ = ν

⇔ ⊆ ⊆

μ ;
- fuzzy complementation is involutive: ( μ C ) C = μ ;
- intersection and union are commutative and associative;
- intersection and union are idempotent and mutually distributive;
- intersection and union are dual with respect to complementation: ( μ

μ ν and ν

ν ) C = ∩

μ C ν C ;

- if we consider that the empty set

∪

∅ is a fuzzy set with an identically zero mem-

bership function, then we have μ ∩∅ = ∅ and μ ∪∅ = μ , for any fuzzy set μ defined

in ;

- if we consider that the universe is a fuzzy set with a membership function equal
to 1, then we have μ

S

∩S = μ and μ ∪S = S , for any fuzzy set μ defined in S .

These properties are the same as the corresponding binary operations. However,
some properties that are true in the binary case are lost in the fuzzy case, such as
the law of the excluded middle ( X

∪

X C =

S

) and the law of non-contradiction

( X ∩ X C = ∅ ). This is because we have, in general:

μ C μ ∪ = S , [8.10]

μ C

μ ∩ = ∅ . [8.11]