This section focused on the main objectives for finding the explicit formulas of and . To attain this, we first determined the numbers and .
4.1. The Number of -Level Fuzzy Subsets
This subsection drove the numbers and in terms of binomial coefficients. To calculate these numbers, we first investigated one of them, e.g., .
We took a set with three elements, e.g.,
The lattice of
, denoted
, 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
(see
Figure 1 and
Figure 2). One could manually determine the number
by describing all chains ordered by the set inclusion
in the following four cases.
- Case I
- Case II
- Case III
- Case IV
The total number
of distinct fuzzy subsets of
was:
The total number of distinct rooted fuzzy subsets of
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.
Lemma 2. For eachand, the numberof all the distinct-level fuzzy subsets ofwas given by the equality: Proof. We started the proof by determining an auxiliary construction. Let
denote the set consisting of all the crisp subsets of
with
-elements,
for
and
. Additionally, the crisp subsets
’s satisfied the following condition:
Next, we aimed to find the cardinality of for , and .
It could be easily proved that
Let
be a chain of the crisp subsets in
for
as follows:
Then, one could obtain the following
-dimensional vector as follows:
For our convenience, we called
the order vector of
. Here, we considered
and for each
,
Then, it was very natural to conclude that:
and
Now, it was easy to find that:
Thus, for the order vector
of
, the number of choices for the first term, the second term,
, and the
k-th term of
were listed as follows:
Therefore, we could precisely write the formula of
as follows:
Hence, the lemma was proved. □
The following result can provide another version of Lemma 2.
Corollary 1. Under the same requirements of Lemma 2, we had: The numbers
’s formed a triangular array and the rows for
to 6 were shown below:
| | | | | | 1 | | | | | | |
| | | | | 2 | | 1 | | | | | |
| | | | 4 | | 5 | | 2 | | | | |
| | | 8 | | 19 | | 18 | | 6 | | | |
| | 16 | | 65 | | 110 | | 84 | | 24 | | |
| 32 | | 211 | | 570 | | 750 | | 480 | | 120 | |
64 | | 665 | | 2702 | | 5460 | | 5880 | | 3240 | | 720 |
The above triangular array was assigned by identity number A038719 in the OEIS and its row sums gave the following sequence (see Theorem 5):
1, 3, 11, 51, 299, 2163, 18,731, 189,171, 2,183,339, 28,349,043, 408,990,251, ⋯.
The elements of the above sequence represent the number of fuzzy subsets of -set, where and was assigned by A007047 in the OEIS.
Remark 3. - 1.
The first three diagonals of the triangular array A038719 gave the following sequences in the OEIS: A000079 (the number of subsets of an-set,, A001047, and A038721 (the number of functionssuch thatcontainedfixed elements,and).
- 2.
The last diagonal and next-to-last diagonal of the triangular array A038719 gave the sequences: A000142 (the number of permutations ofletters,,or factorial numbers) and A038720, respectively, in the OEIS.
The same methods as that in Lemma 2 were exemplified in the following two lemmas and two corollaries to derive the formulas of and .
Lemma 3. For each natural numberand, the numberof all the distinct-rooted-level fuzzy subsets ofwas given by the equality:withfor.
Corollary 2. Under the same assumptions of Lemma 3, we had: Lemma 4. For any nonzero positive integersand k,the numberof all the distinct-rooted-level fuzzy subsets of was given by the equality:withfor.
Corollary 3. Under the same assumptions of Lemma 4, we had: The numbers satisfied by the formula of
formed a triangle and the first few rows of the triangle were listed below:
| | | | | | 1 | | | | | | |
| | | | | 1 | | 1 | | | | | |
| | | | 1 | | 3 | | 2 | | | | |
| | | 1 | | 7 | | 12 | | 6 | | | |
| | 1 | | 15 | | 50 | | 60 | | 24 | | |
| 1 | | 31 | | 180 | | 390 | | 360 | | 120 | |
1 | | 63 | | 602 | | 2100 | | 3360 | | 2520 | | 720 |
The above triangle was assigned by A028246 in the OEIS and its row sums gave the following sequence (see Theorems 6 and 7):
1, 2, 6, 26, 150, 1082, 9366, 94,586, 1,091,670, 14,174,522, 204,495,126, 3,245,265,146,.
The elements of this sequence correspond to the number of rooted fuzzy subsets of -set, where and was assigned by the identity number A000629 in the OEIS.
Remark 4. - 1.
The first four diagonals of the triangle A028246 gave the following sequences in the OEIS: A000012 (the all one′s sequence), A000225 (the number of nonempty subsets of-set, ), A028243 (the number of functionssuch thatcontained two fixed elements,and), and A028244 (the number of functions, such thatcontainedfixed elements,and).
- 2.
The last seven external diagonals of the triangle A028246 gave the following sequences in the OEIS: A000142 (the number of permutations of letters,) and A001710 (number of even permutations of letters, ), A005460 (essentially Stirling numbers of the second kind, offset: 0,2), A005461 (essentially Stirling numbers of the second kind, offset: 1,2), A005462 (essentially Stirling numbers of the second kind, offset: 3,2), A005463 (essentially Stirling numbers of the second kind, offset: 4,2), and A005464 (essentially Stirling numbers of the second kind, offset: 5,2), respectively, in the OEIS.
By taking in the explicit expression of in Lemma 2, we had an exciting result.
Corollary 4. The numberof all the maximal chains of the crisp subsets ofwas given by the equality: Proof. The proof directly followed from the explicit formula of by substituting
Corollary 5. The numbersandof all the maximal chains of the crisp subsets ofwere derived by the equality: Proof. It could be proved similarly as in Corollary 4, by using Lemmas 3 and 4. □
Starting with , the first few terms of the factorial of were:
.
This sequence was assigned by A000142 in the OEIS.
The formulas of , , and could also be justified by using Corollaries 1–3, respectively. To highlight and verify the above results, we illustrated them by the following example.
Example 4. Determine the numbers,, and.
Solution. One could calculate the numbers
,
, and
in the following, by substituting
and
in Lemmas 2–4, respectively. We had:
Hence, the formulas were verified.
As a verification of the above Corollaries 4 and 5, we manually computed all the maximal chains of the crisp subsets of
. We listed all the maximal chains,
, as follows:
These 24 maximal chains agreed with the numbers , and .
We could have analyzed the time complexity of calculating as follows: The calculation of , needed . needed operations of the above calculation and multiplied them. Thus, its time complexity was , which was .
4.2. Explicit Formulas of , and
This subsection derived the explicit summation formulas of , and . Now, we gave the most important result in the following theorem.
Theorem 5. The numberof all the distinct fuzzy subsets ofwas given by the equality:and where
is arbitrary. Proof. By summing up all ’s in Lemma 2 and Corollary 1, we had the required formula for . □
For
, the formula
gave the following sequence with the initial terms (See
Table 2):
This sequence was assigned by A007047 in the OEIS.
One could have two immediate straightforward consequences of Theorem 5 in the following:
Theorem 6. For a fixed valueand, the numberof all the distinct-rooted fuzzy subsets ofwas given by the equality:andwith.
Theorem 7. The numberof all the distinct-rooted fuzzy subsets ofwithwas given by the equality:andwith.
The beginning for
to
, the results of theormula
were shown below (see
Table 3):
This sequence was assigned by A000629 in the OEIS.
The following observations gave the connections among the numbers of
, and
. It was similar to Proposition 1 in [
32] and Theorem 1 in [
33].
Proposition 3. For any positive integer, we had: To highlight and verify the above results, we illustrated it through an example in the following (see
Figure 3):
Example 5. Find the numbers, and.
Solution. By taking
in the explicit formulas of
, and
, we had
, and
in the following way:
and
We could have analyzed the time complexity of directly calculating as follows. The maximum time complexity for computing a binomial coefficient was . For a )-tuple sequence satisfying , the time complexity for calculating was, thus, . However, there were sequences in; thus, the time complexity for was not polynomial.