We know that the cardinality of a finite crisp (or, classical) set $A$ can be considered as a measure of "number of elements" of $A$. However, if $X$ is a universe of discourse and $\tilde A$ (or, simply $A$) is a Fuzzy Set (FS) defined on $X$ such that $$A=\{(x,\mu_A(x))\,:\,x\in X\}$$ where $\mu_A:X\to[0,1]$ is the membership function of the fuzzy set $A.$ Then the scalar-cardinality or the sigma-count ($\Sigma$-count) of the fuzzy set $A$ is defined as (see here): $$\lvert A\rvert= \begin{cases} \displaystyle\sum_{x\in X} \mu_A(x),&\text{if }A\text{ is a discrete FS} \\ \displaystyle\int_{x\in X} \mu_A(x)\,dx,&\text{if }A\text{ is a continuous FS.} \end{cases} $$ It is said to be an general extension of the notion of cardinality of a set!
My question is:
What does it even mean in case of fuzzy sets?
(I mean we are just adding the membership grades of $x$!!)
Again, if $A$ is an Intuitionistic Fuzzy Set (IFS) on $X,$ $A$ is defined as: $$A=\{(x,\mu_A(x),\nu_A(x))\,|\,x\in X\}$$ where $\mu_A(x)$ denotes the membership grade of $x$ in $A$ and $\nu_A(x)$ denotes the non-membership grade of $x$ in $A.$ Furthermore, both $\mu_A(x)$ and $\nu_A(x)$ lie in $[0,1]$ such that $0<\mu_A(x)+\nu_A(x)\le1,\forall x\in X.$
The scalar-cardinality or the $\Sigma$-count of the IFS $A$ is defined as (as can be seen here): $$\Sigma\text{-count}(A)=\left[\sum_{x\in X}\mu_A(x),\sum_{x\in X}(1-\nu_A(x))\right].$$ Question:
What does it signify?
Then there are Neutrosophic Set (NS) & Neutrosophic Fuzzy Set (NFS), which arise in Neutrosophy/Neutrosophic logic (e.g., see here), which can be considered as a generalization of an IFS.
Now, if $A$ is a Single Valued Neutrosophic Fuzzy Set (SVNFS) on $X,$ then $$\text{SVNFS } A=\{\langle x,\mu_A(x),T_A(x),I_A(x),F_A(x)\rangle\,:\,x\in X\}$$ where all of $\mu_A,T_A,I_A$ and $F_A$ are functions from $X$ into $[0,1],$ and denote respectively the membership grade, the truth membership grade, the indeterminacy membership grade and the falsity membership grade of $x$ in $A.$ Moreover, $\mu_A(x),T_A(x),I_A(x)$ and $F_A(x)$ are independent measures for each $x\in X,$ and $0\le T_A(x)+I_A(x)+F_A(x)\le3,\forall x\in X.$
Question:
Does the term scalar-cardinality make sense in case of SVNFSs as before? If yes, how could one define the same?
Please give some insights...