On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences

Throughout this paper, we assumed $X$ to be a nonempty finite set consisting of $n$ elements, where $n$ is a nonzero positive integer. A fuzzy subset $\tilde{A}$ of the set $X$ is defined by a membership function $\tilde{A}:X\to I,$where $I=\left[0, 1\right].$The set of all fuzzy subsets of $X$ is denoted by $I^X$, fuzzy subsets by $\tilde{A},\tilde{B},\tilde{C},$ etc., and its membership values by $\alpha ,\beta ,\gamma ,$ etc. The union $\left({\displaystyle \cup }\right)$ and intersection $\left({\displaystyle \cap }\right)$ of two fuzzy subsets were defined by using supremum and infimum pointwise, respectively, and the complementation ${(}^c)$ of a fuzzy subset by $1-\tilde{A}$ operator pointwise [22,24]. A fuzzy subset $\tilde{A}$ was stated to be contained in a fuzzy subset $\tilde{B}$, denoted by $\tilde{A}\subseteq \tilde{B},$ if $\tilde{A}\left(x\right)\le \tilde{B}\left(x\right),\forall x\in X$. A fuzzy subset $\tilde{A}$ was stated to be contained in a fuzzy subset $\tilde{B}$, denoted by $\tilde{A}\subseteq \tilde{B},$ if $\tilde{A}\left(x\right)\le \tilde{B}\left(x\right),\forall x\in X$. Through the support and core of a fuzzy subset of $\tilde{A}$, we mean the crisp subsets $Supp \left(\tilde{A}\right)=\{x\in X:\tilde{A}\left(x\right)>0\}$ and $Core \left(\tilde{A}\right)=\left\{x\in X:\tilde{A}\left(x\right)=0\right\}$, respectively. When stating the image of a fuzzy set $\tilde{A}$, we mean the set of membership values of $\tilde{A}$ denoted by$Im(\tilde{A})=\tilde{A}\left(X\right)$. Through an $\alpha$-cut [22,23] of a fuzzy subset $\tilde{A}$ for $\alpha$ belongs to $I,$ we had a crisp subset ${\tilde{A}}^{\alpha }=\left\{x\in X:\tilde{A}\left(x\right)\ge \alpha \right\}$ of $X$. This is called the weak $\alpha$-cuts. By strong $\alpha$-cuts, we mean ${\tilde{A}}_{\alpha }=\{x\in X:\tilde{A}\left(x\right)>\alpha \}$. In this paper, we were always dealt with weak $\alpha$-cuts. It was easy to verify that for $0\le \alpha \le \beta \le 1$, we had ${\tilde{A}}^{\beta }\subseteq {\tilde{A}}^{\alpha }$. For any arbitrary fuzzy subset $\tilde{A}$, we has a result as follows.

, where ${\chi }_{{\tilde{A}}^{\alpha }}$ denotes the characteristic function of the crisp subset ${\tilde{A}}^{\alpha }$.

This subsection presents some known definitions and results in the area of combinatorics to deal with our main results.

We displayed some of its basic properties as follows:

The fourth item is the most important recurrence relation for binomial numbers.

We provided a familiar formula for binomial numbers as follows (by definition, $0!=1$) (See Table 1):

Next, we provided a famous theorem, known as the Binomial Theorem.

The binomial numbers can be put into a triangular array, which is called Pascal′s triangle. Pascal′s triangle was assigned by A007318 in the OEIS.

										1
									1		1
								1		2		1
							1		3		3		1
						1		4		6		4		1
					1		5		10		10		5		1
				1		6		15		20		15		6		1
			1		7		21		35		35		21		7		1
		1		8		28		56		70		56		28		8		1
	1		9		36		84		126		126		84		36		9		1

The row sums of Pascal′s triangle provided the sequence:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16,384, 32,768, 65,536,$\cdots$.

The elements of this sequence represented the number of subsets of an $n$-set, $n\in {\mathbb{N}}^{*}=\mathbb{N}{\displaystyle \cup }\left\{0\right\}$. This sequence was also assigned by A000079 in the OEIS.

This section discusses the work related to the counting of fuzzy subsets. For more concepts, the reader can refer to [27,28,29].

In [28,30], Murali and Makamba introduced an equivalence relation ($\cong$) on $I^X$ by defining $\tilde{A}\cong \tilde{B}$ if $\tilde{A}\left(x\right)>\tilde{A}\left(y\right)\iff \tilde{B}\left(x\right)>\tilde{B}\left(y\right),\tilde{A}\left(x\right)=0\iff \tilde{B}\left(x\right)=0$, and $\tilde{A}\left(x\right)=1\iff \tilde{B}\left(x\right)=1$ for all $x,y\in X$. This equivalence relation agreed with the equality of crisp subsets when the binary values $\left\{0,1\right\}$ were used.

A fuzzy subset was represented by a pinned flag $\left(\mathfrak{F},\ell \right)$ through $\alpha$-cuts, which was written as follows: where $\mathfrak{F}$ is a flag on $X$, that is a chain $X_0\subset X_1\subset X_2\subset \cdots \subset X_n=X$ called the maximal chain (flag) of crisp subsets of $X,$ and $\ell$ is a keychain from $I$, i.e., $1\ge {\lambda }_0\ge {\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n\ge 0$ [30,31].

Let $F_n$ denote the number of distinct fuzzy subsets of $X$ over $\cong$. Through the detailed study of pinned flags, an explicit formula of $F_n$ was attained by accumulating all integer partitions $\tau ⊢n$. That is, $\tau ⊢n$ means that τ is a partition of the integer $n$.

The formula as mentioned above is called the factorial summation formula for $F_n$. In [10], the author also discussed some different methods of counting the number of fuzzy subsets. This section presents some of the main results to focus our concepts.

A closed-form of $F_n$ was expressed by the following theorem.

The expression as the infinite series for $F_n$ was given below.

Since permutations were taken, a significant role in the computation of an exponential generating function $F_n$, the function $F_n$ was derived in [10] as follows:

The idea of this subsection was to briefly discuss an equivalence relation $(\cong )$ on the set I^X. It was defined in the following way:

Indeed, one can verify that this relation is an equivalence relation on $I^X$. This equivalence relation coincides with the equality of crisp subsets when the values of membership in $I=\left[0,1\right]$ are restricted to binary values $I^{\prime }=\left\{0,1\right\}$. Based on this equivalence relation, the equivalence class having $\tilde{A}$ was denoted by $\left[\tilde{A}\right]$. We stated that two fuzzy subsets $\tilde{A}$ and $\tilde{B}$ were distinct if $\tilde{A}$ and $\tilde{B}$ were not equivalent, that is, $\tilde{A}\ncong \tilde{B}$.

The motivation behind this subsection was to concentrate on classifying fuzzy subsets of $X$ using the idea of $k$-level fuzzy subsets. Now, we introduced some definitions to derive the main results of this paper in the next section.

One can see that for $\alpha \in I$ and $1\le k\le n$, $\alpha$-cuts of a $k$-level fuzzy subset $\tilde{A}$ of $X$ formed a chain of $\left(k+1\right)$-pair of crisp subsets of length k ordered by an usual set inclusion as follows: with:written in the magnitude of the descending order, not necessarily having $0$ and $1$.

That is, a fuzzy subset $\tilde{A}, \tilde{A}=\underset{{\alpha }_i}{{\displaystyle \cup }}\left\{{\alpha }_i{\chi }_{{\tilde{A}}^{{\alpha }_i}}:0\le {\alpha }_i\le 1\right\}$ had its ${\alpha }_i$-cuts equal to ${\tilde{A}}^{{\alpha }_i}$’s. Here, ${\tilde{A}}^{{\alpha }_i}$’s were various components of the above chain $\Lambda$. We stated that two $k$-level fuzzy subsets were distinct if they were not equivalent. Therefore, it could be concluded that $\tilde{A}\cong \tilde{B}$ if, and only if, $\tilde{A}$ and $\tilde{B}$ had the same chain of crisp subsets of type $\Lambda$.

A fuzzy subset of $X$ having $n$ elements could have maximum $n$-distinct membership values in $I$, and then the maximum number of distinct $\alpha$-cuts of a fuzzy subset of $X$ would be $n+1$. For $k=n$ in the above chain $\mathsf{\Lambda }$, one could obtain a maximal chain, which is called a flag. Therefore, the length of the flag for a fuzzy subset of $X$ is $n$.

For the sake of understanding Definition 4, let us illustrate the definition through the following example:

From the above discussions, it can be followed that there exists a bijection between the collection of all distinct $k$-level fuzzy subsets of $X$ and the collection of all chains of crisp of subsets of $X$ of length $k$ ordered by the set inclusion. Similarly, there was a bijection between the collection of all distinct $\varnothing$-rooted (respectively, $X$-rooted) $k$-level fuzzy subsets of $X$ and the set of all $\varnothing$-rooted (respectively, $X$-rooted) chains of crisp of subsets of $X$ of length $k$ ordered by the set inclusion, respectively.

Note that if $\Gamma$ was a chain such that its initial term and the terminal term coincided, then $\Gamma$ was a chain of length $0$.

That is, ${ℱ}_{n,k}, {ℱ}_{n,k}^{\varnothing }$ and ${ℱ}_{n,k}^X$ were the collection of all chains of crisp subsets of $X$ of length $k$, the collection of all $\varnothing$-rooted chains of crisp subsets of $X$ of length $k,$ and the collection of all $X$-rooted chains of crisp subsets of $X$ of length $k$ ordered by the set inclusion, respectively. We used notations $F_{n,k},F_{n,k}^{\varnothing },$ and $F_{n.k}^X$ to denote the cardinal numbers of ${ℱ}_{n,k},{ℱ}_{n,k}^{\varnothing },$ and ${ℱ}_{n,k}^X$, respectively. Further we define $F_{0,0}=1, F_{0,0}^{\varnothing }=1$ and $F_{0,0}^X=1$.

Let ${ℱ}_n,{ℱ}_n^{\varnothing },$ and ${ℱ}_n^X$ be the set of all chains of crisp subsets of $X$, the set of all $\varnothing$-rooted chains of crisp subsets of $X$, and the set of all $X$-rooted chains of crisp subsets of $X$ ordered by the set inclusion, respectively. Take $F_n=\left|{ℱ}_n\right|,F_n^{\varnothing }=\left|{ℱ}_n^{\varnothing }\right|,$ and $F_n^X=\left|{ℱ}_n^X\right|$. Assume that $F_0=1,F_0^{\varnothing }=1,$ and $F_0^X=1$ for the counting formulas of $F_n,F_n^{\varnothing },$ and $F_n^X$ for $n=0$, respectively, as a fuzzy subset of the empty set has no conventional meaning. It can also be concluded that there also exists a bijection between the collection of all distinct fuzzy subsets of $X$ (respectively, distinct $\varnothing$-rooted fuzzy subsets of $X$, distinct $X$-rooted fuzzy subsets of $X$), and the collection of all chains of crisp subsets of $X$ (respectively, $\varnothing$-rooted chains of crisp subsets of $X,$ $X$-rooted chains of crisp subsets of $X$) ordered by the set inclusion.

This subsection drove the numbers $F_{n,k},F_{n,k}^{\varnothing },$ and $F_{n.k}^X$ in terms of binomial coefficients. To calculate these numbers, we first investigated one of them, e.g., $F_{n,k}$.

We took a set with three elements, e.g., $X_3=\left\{1, 2, 3\right\}.$ The lattice of $\mathcal{P}\left(X_3\right)$, denoted $L\left(\mathcal{P}\left(X_3\right)\right)$, was generated by the following subsets (See Figure 1):

It was essential to list these subsets because they were the key point in drawing a lattice diagram and for counting the number of chains of crisp subsets of $X_3$ (see Figure 1 and Figure 2). One could manually determine the number $F_3$ by describing all chains ordered by the set inclusion $(\subset )$ in the following four cases.

The total number $\left|{ℱ}_3\right|$ of distinct fuzzy subsets of $X_3$ was:

The total number of distinct rooted fuzzy subsets of $X_3$was:

By examining the above discussions and analyzing Figure 1, one could conclude that the calculations manually consumed a lot of time, which is a difficult task. Therefore, it implied that a new technique was essential to solve these problems.

Now,

Next, we aimed to find the cardinality of ${ℱ}_{n,k}$ for $k=0,1,2,\cdots ,n$, and $n\in \mathbb{N}$.

It could be easily proved that $F_{n,0}=\left|{ℱ}_{n,0}\right|=2^n.$ Let $\Gamma$ be a chain of the crisp subsets in ${ℱ}_{n,k}$ for $k=0,1,2,\cdots ,n$ as follows:

Then, one could obtain the following $\left(k+1\right)$-dimensional vector as follows:

For our convenience, we called $\alpha$ the order vector of $\Gamma$. Here, we considered and for each $\alpha \in \Omega$,

Then, it was very natural to conclude that: and