Fuzzy subset

Some set-theoretical notions for fuzzy subsets

In classical mathematics the definitions of union, intersection and complement are related with the interpretation of the basic logical connectives $\vee ,\wedge ,\neg$. In order to define the same operations for fuzzy subsets, we have to fix suitable operations $\oplus ,\otimes$ and ~ in L to interpret these connectives. Once this was done, we can set

$(s\cup t)(x)=s(x)\oplus t(x)$,

$(s\cap t)(x)=s(x)\otimes t(x)$,

$(-s)(x)=~s(x)$.

Also, the inclusion relation is defined by setting

$s\subseteq t\Leftrightarrow s(x)\leq t(x)$ for every $x\in S$.

In such a way an algebraic structure $([L^{S},\cup ,\cap ,-,\emptyset ,S)$ is defined and this structure is the direct power of the structure $(L,\oplus ,\otimes ,$~,0,1) with index set S. In Zadeh's original papers the operations $\oplus ,\otimes$, ~ are defined by setting for every x and y in [0,1]:

$x\otimes y$ = min(x, y) ; $x\oplus y$ = max(x,y) ; $~x$ = 1-x.

In such a case $([0,1]^{F},\cup ,\cap ,-,\emptyset ,S)$ is a complete, completely distributive lattice with an involution. Several authors prefer to consider different operations, as an example to assume that $\otimes$ is a triangular norm in [0,1] and that $\oplus$ is the corresponding triangular co-norm.

In all the cases the interpretation of a logical connective is conservative in the sense that its restriction to {0,1} coincides with the classical one. This entails that the map associating any subset X of a set S with the related characteristic function is an embedding of the Boolean algebra $({0,1}^{S},\cup ,\cap ,-,\emptyset ,S)$ into the algebra $(L^{S},\cup ,\cap ,-,\emptyset ,S)$.