Is there a definition for "equality of fuzzy sets" ?
My current thinking :
Say we have two fuzzy sets $A = \{(x,\mu_{A}(x)):x \in X\}$ and $B = \{(y,\mu_{B}(y)):y \in Y\}$
When we consider equality of fuzzy sets , then for making $A = B$ , we might want :
- (I) $X = Y$
- (II) $\mu_{A}(x) = \mu_{B}(x)$
Are these 2 conditions (I) & (II) necessary ?
Alternately , we can consider like this :
- (III) $support(A) = support(B)$
- (IV) $x \in support(A) \Rightarrow \mu_{A}(x) = \mu_{B}(x)$
Are these 2 conditions (III) & (IV) sufficient ?
A simple example for clarifying my question:
Are two fuzzy sets $A$ and $B$ with the following definitions equal?
$$ A = 0.1/0 $$ $$ B = 0.1/0 + 0/1 $$
According to Jon Y. Kim , "Introduction to Fuzzy Set Theory" , Equality of Fuzzy Sets can be achieved easily , though the Initial Definition of Fuzzy Set itself requires a little tweaking like this :
Whereas OP has used Collection $X$ in Fuzzy Set $A$ & Collection $Y$ in Fuzzy Set $B$ , it seems to be more standard to have Single Universal Collection , for which I will use $U$ here , to avoid confusion with the other existing letters.
Each Fuzzy Set then uses Collection $U$ and will have the membership function $\mu$.
Definition (Fuzzy Set) If $U$ is a collection of objects denoted generically by $u$, then a fuzzy set $A$ in $X$ is a set of ordered pairs :
$A = \{(u, \mu_A(u)) : u \in U\}$
where $\mu_A(u)$ is the membership function which maps $u$ to a real number in the interval $[0, 1]$
Then $A=\{(u,\mu_A(u)):u \in U\}$ and $B=\{(u,\mu_B(u)):u \in U\}$
We no longer require OP Condition (I) , which is automatically true.
[[ When OP had $X$ & $Y$ not equal , in general , then $U$ will contain at least $X \cup Y$ , with suitable $\mu=0$ to exclude the unwanted elements are inserted between $X,Y,U$ ]]
Now , Kim Jon makes the Equality Definition like this :
Definition (Empty Fuzzy Set) A fuzzy set $A$ is empty if and only if its membership function is identically zero on $X$, $\mu_A \equiv 0$
Definition (Fuzzy Set Equality) Two fuzzy sets $A$ and $B$ are equal if and only if $\mu_A(x) = \mu_B(x)$ for all $x ∈ X$. We will denote equality as $A = B$
This matches OP Condition (II) , which is necessary & sufficient here.
Moving to OP Condition (III) , we see that it is necessary , but not sufficient. When Supports are Equal , we can not say Fuzzy Sets are Equal , because the membership functions might not match.
We then include OP Condition (IV) , which says that membership functions are equal within the Common Support.
Now , Fuzzy Set Equality is achieved , but (III) is redundant.
We know that elements outside the Support have $\mu_A(u)=0=\mu_B(u)$ , thus , it is Equal within & outside the Support , throughout Common Universal Set $U$ !
Thus (IV) is enough , provided we use the Fuzzy Set Definition with Common Universal Set $U$.
When we do not use Common Universal Set $U$ , then (III) + (IV) are sufficient too.
Kim Jon uses Concepts from Zadeh & Zimmerman , hence it is somewhat standardized.
EXAMPLE :
$A = 0.1/0$
$B = 0.1/0 + 0/1$
We have to write it out with Common Universal Collection $U$ , which should contain at least the Union of the 2 Supports.
Let $U=\{0,1,2,3\}$
$A = 0.1/0 + 0.0/1 + 0.0/2 + 0.0/3$
$B = 0.1/0 + 0.0/1 + 0.0/2 + 0.0/3$
$C = 0.1/0 + 0.0/1 + 0.0/2 + 1.0/3$
$D = 0.0/0 + 0.0/1 + 0.0/2 + 0.0/3$
Now (I) is automatically true.
More-over (II) is also true.
Hence we have $A=B$ here.
It is not true that $A=C$ & $B=C$ , because membership function will not match for at least 1 element.
Like-wise , $A=D$ & $B=D$ & $C=D$ can not be true.
Lastly "D = Empty Fuzzy Set" is true.