An abstract approach to sequences and series using soft complex numbers

Abstract

Keywords: Soft Cauchy Sequence; Soft Complex Convergence; Soft Complex Number; Soft Complex Sequence; Soft Complex Series; Soft Set

Introduction

An adaptable scheme for handling ambiguity and uncertainty in a variety of everyday circumstances is provided by fuzzy set [1]. One benefit is that it may depict the differentiation between membership and non-membership, enabling more complex conceptual depictions. Because of its versatility, complex systems with ill-defined constraints can be better modeled. This theory has successfully been applied in various fields of study like differential equations [2–5], and decision-making [6,7]. To equip fuzzy

set with parameterization mode, Molodtsov [8] introduced a groundbreaking mathematical approach known as soft set theory (SST), designed to address uncertainties. It has multiple applications in the visualization of knowledge, data mining, and governance. Its capacity to manage the confusion and unpredictability present in practical problem data is a significant benefit. A more adaptable and truthful depiction of intricate systems is made possible by soft sets (SSETs), which permit the inclusion of elements with incomplete memberships. In decision support systems,

*Corresponding author. *E-mail address: aurkhb@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey © Author. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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where precise data may be omitted or it may be challenging to collect, adaptability is especially essential. Subsequent research by Maji et al. [9,10] expanded the scope of SST, presenting various operations applied to decision-making problems. Rahman et al. [11] employed parameterized approach to decision making using extension of fuzzy SSETs. Vimala et al. [12] investigated some algebraic structures and their relating results using multi aspects of fuzzy SSETs. Further contributions by Shabir and Naz [13] introduced soft topological spaces, while Majumdar and Samanta [14] explored mappings between SSETs. The study by Feng et al. [15] investigated the intersection of SSETs with fuzzy and rough sets. Researchers in [16,17] introduced soft real sets (SRSs), soft real numbers (SRNs), soft complex sets (SCSs), and soft complex numbers (SCNs), demonstrating their properties and practical applications. Saeed et al. [18] introduced the notions of soft elements and members. Later on, they [19] introduced the notions of soft algebraic structures using soft elements and members. A subsequent study in 2016, focused on fundamental operations in a soft context, discussing soft functions and including results like the Bolzano theorem [20]. Irfan [21] studied various properties of convexity and its inverse in soft and fuzzy SSETs environments. Salih and Sabir [22] explored specific characteristics related to convexity and concavity within the context of SSETs. The researchers [23–27] extended the previously defined knowledge of convexity to some of its other variants in soft and fuzzy SSET environments. Further study related to ambiguous focus on fuzzy complex numbers [28] that extend traditional complex numbers to accommodate uncertainty and imprecision, enabling degrees of membership and uncertainty in both real and imaginary components. Demir [29] discussed the idea of soft complex valued metric spaces and looked at a few topological facets of them. Next, for a variety of soft mappings on soft complex valued metric spaces, certain fixed soft element theorems were developed. Selvachandran et al. [30,31] studied the relations and distance measures between complex vague SSETs. Sabir [32] studied the extension of complex fuzzy set. This extension enhances the representation of fuzzy information within mathematical contexts, particularly in systems where imprecision and ambiguity play a significant role. Wardowski [33] demonstrated the natural relationship between soft operations and soft objects in soft topological spaces and proposed a novel concept of soft element of a SSET. Öztürk [34] examined the connection between bipolar SSETs and bipolar soft points and originated the idea of bipolar soft points. A novel interpretation of soft point was introduced by Senel [35], which allows one to construct all soft points that vary with any parameter that occurs in a SSET. Taşköprü and Altntaş [36] provided a topology whose members are sets of soft elements and explained how this topology relates to basic soft topology. Allam et al. [37] first proposed the idea of a soft element and then modified the idea of a soft point to eliminate any connected issues with the soft point. Polat et al. [38] introduced a number of

properties on soft topological spaces, based on the notion of soft elements, which provide us with an alternative viewpoint for the advancement of soft set theory. These qualities include the neighbourhood structure of a soft element, soft interior, and among others. Hameed and Khalil [39] proved some new theorems on the equality of infimum soft sequences. Asma et al. [40] and Kirişçi [41] discussed decision making applications based on uncertain environments. Novelty and Motivation The scientific literature reviewed above makes it evident that while certain scholars have studied soft elements, soft members, and soft complex numbers, there needs to be moreresearch on the description of soft complex sequences and series in the literature. The SCNs provide an effective tool for handling equivocal and imperfect data in an organized way. Their capacity to express indeterminate values withreal and imaginary aspects is one of its advantagessince it enables a more thorough explanation of intricate situations. Because of its adaptability, it may be utilized for analyzing systems with multifaceted unpredictability, which is prevalent in a wide range of practical uses. Furthermore, while maintaining uncertain knowledge, SCNs offer an adaptable structure for carrying out computations. The SCNs and soft complex sequence (SCSQs) are useful in many domains where handling ambiguous or imprecise data is crucial. They can be employed to simulate systems with ambiguous or fuzzy parameters in engineering, enabling more reliable and flexible control schemes. The SCNs, which take into account erratic characteristics and unclear market data, can help in risk evaluation and investment optimization in the financial sector. The SCSQs can also be used in medical diagnosis to examine patient information as well as medical records, taking into account the inherent uncertainty in diagnosis and treatment results. Additionally, SCNs can also improve the processing of ambiguous or partial data in artificial intelligence and machine learning, resulting in more trustworthy and precise decisions. All things considered, SCNs have implications in a variety of domains, such as financial services, engineering, and artificial intelligence, where managing uncertainty is essential to solving problems successfully. Thus, keeping in mind above mentioned advantages and adaptability of SCNs, this study is aimed to introduce the novel concept of a SCSQ, examining its convergent components using SCNs and soft real sequences (SRSQ). Additionally, it establishes a convergent series on SCNs, exploring their relationship. Section 2 reviews some basic concepts related to soft sets, soft elements, soft members, soft complex numbers, etc., and this is how the rest of the article is structured. In Section 3, the soft complex limit and associated theorems have been characterized and described using SCNs and SCSQs. Section 4 has a discussion of the boundedness of SCSQs, soft complex Cauchy sequences using SCNs, and their corresponding results. In Section 5, the convergence of these SCNs soft complex series and the corresponding

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outcomes have been examined. Lastly, Section 6 presents the overview, future scope, and implications of the anticipated effort

Preliminary Knowledge

The following section is meant to review fundamental definitions and operations for SSETs, SRSs, and SCSs, as outlined in [13,16,17,42]. Definition 1 Let be an introduction to the universe of discussion and be a set of parameters. Let shows the set of power of and be a subset that is not empty of . A pair ( , ) is called a SSET over , where is a mapping given by . Put otherwise, a set above is a family of parameterized discourse universe subsets . For might be seen as a set of approximation SSETs elements ( , ) and if then , that is : , . The family of all these SSETs over denoted by . Definition 2 Let , then is a soft subset of , symbolized by , if and . In this case is described as a soft subset of , and is described as a soft super set, , . Definition 3 A pair of soft subsets and over are considered equal if is a soft subset of and is a soft subset of . Definition 4 The soft subsets complement ( , ) denoted by ( , )c is determined by ( , )c = ( , ): is a mapping provided by and is said the soft complement function is equivalent to . Clearly ( ) is similar to and . Definition 5 The difference between SSETs ( , ) and ( , ) over displayed by ( , ) - ( , ) is the SSET (D, ) such that . Definition 6 The union of SSETs and ( , ) Over is the SSET ( ), where, for all ,

where, and and written as . Definition 7 The intersection ( ) between two SSETs ( , ) and ( , ) over , denoted , ( , ) and ( , ) can be defined as and . Definition 8 Given a non-empty parameter set and a non-empty set . Then a function is defined as a soft element in . A soft element τ of belongs to a SSET ( , ) of which is symbolized by if . Thus for a SSET ( , )of concerning the index set , it gives . In that case, τ is said to be a soft element of the SSET ( , ).

the For SSETs ( , ) and ( , ). ( , ) within relation . holds iff each soft element belonging to ( , ) is also a soft element in ( , ). Definition 9 Consider R as the collection of real numbers, let 2(R) be the set of all bounded, non-empty subsets of R, and be a set of parameters. In this context, a mapping is termed a SRS, denoted as ( , ). In particular if ( , ) is a singleton, then recognizing ( , ) together with the equivalent soft element, it will be defined as a SRN. Definition 10 The sequence it is claimed that a set of SRNs is convergent to the soft real limit in R( ), when it has the soft real limit , otherwise it is claimed as divergent. Lemma 1 [42] There is a soft convergent sub sequence for every SRSQ. This outcome confirms clearly that there is a convergent sub sequence for any bounded SRSQ. Definition 11 Let C denote the collection of complex numbers, Λ (C) be the collection of all bounded non-empty subsets of the complex number set and be a set of parameters. Then a mapping is said to be a SCS. It’s shown by ( , ). Definition 12 In particular, if ( , ) a SSET with a singleton, then identifying ( , )it will be defined as a SCN together with the corresponding soft element. We used the symbols and , to denote SCNs. Example 1 Let C be the set of complex numbers. The be a set of parameters given by . Then ( , ) can be considered as a SCN where

Definition 13 A SCS ( , ) is claimed as soft pure imaginary set if is a subset of the set of pure imaginary numbers for each . Let us denote the set of all SCSs by and the set of all SCNs by . Definition 14 Assume that is a SCS or number. Subsequently, Re is a real part and lm is an imaginary part of and are stated as follows: (2) and (3) . It is shown that Re and lm are SRSs For every or numbers. Definition 15 Let ( , ) be a collection of SCS. Then the modulus of ( , ) is symbolized by ( , ) and is defined by . Limits of a Sequence in SCNs and be any two SRSQs then Definition 16 If is a SCSQ denoted by

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Mechanism for the characterization of SCSQs can be viewed from Figure 1. of SCNs is called conDefinition 17 The sequence , when it has the soft vergent to soft complex limit complex limit , otherwise it is claimed as divergent. represents a sequence of SCNs. Definition 18 If We conclude that approaches the soft complex limit (as ) whenever for every SRN , a soft positive integer (SPI) depending on such that

and . We now prove Proof. Suppose . Indeed, fix and find, by the stated that . It given that convergence, there exists such that and . Then, for , it gives

Conversely, suppose that . Then, given , it holds that we find SPI such that for . So clearly

Then, we claim that is the number of . In fact, the soft complex, where it gives that . Since is a given finite set of parameters. Because of the Archimedean char, such that . Given acteristic of real numbers, the fact that

and be convergent infinite Proposition 1 Let , sequences of SCNs. Then the sequences , and are convergent, and

In addition, if then the sequence for every and . It means that . be a sequence in the SCNs. Write Theorem 1 Let where and are sequences of SRNs. iff and Then the given sequence convergent in converge in R ( ).

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limits sequence that there are SPI and depending on such that whenever and whenever . If , where is the maximum of and , then

be some complex number. We prove that . Now, given any SPI , we can choose a soft positive small enough to ensure that . Now , and therefore, there exists some SPI Let

. This gives the fact that . , we see that On apply this result with . It follows that . Next we prove that if and are infinite sequences of SCNs, and if and , then . Suppose some SRN be given, it follows from the definition of limits sequence that there exists SPSIs and such that whenever and whenever . Suppose be the maximum of and , if then that and . Therefore, . It shows that . and

We can apply this result with for all positive integer n, where and . Now and that

for every positive inteFinally, suppose that gers n, and that . Then for any given SRN , there exists some SPIs such that whenever

Theorem 2 If is the convergent soft complex limit for the SCSQ , then it is also the convergent soft complex . limit for converges to soft complex limit Proof. Since then for any SRN , there exists SPI such , for all . If we take m = n + s, that , then implies m = n + s and for all . Hence is the convergent soft complex . limit for the SCSQ Theorem 3 A convergent sequence of SCNs has a unique soft complex limit. be a soft complex convergent sequence. Proof. Let and . We wish to Suppose that . For given , by the Definition 18, there prove implies exists a soft real positive integer such that . Moreover, there exists a soft real positive such that implies . For integer , by the triangle inequality it gives

This shows that for all . It follows that and hence, . is convergent iff the SRSQ Theorem 4 The SCSQ is also convergent. be a soft complex convergent sequence, Proof. Let . For given , by Definition 18, there and let implies exists a soft real positive integer such that . So by the triangle inequality we can show . Conversely, can be shown that similarly. converges to zero iff the Remark 1 The SCSQ converges to zero. SRSQ Bounded by the Sequence of SCNs is Definition 19 Soft complex number sequence exists such that claimed as bounded if a SRN . A sequence is unbounded if it is not bounded. Example 3 Consider is a finite set of parameters. Then, it gives where that

be a SCSQ. We say that is Definition 20 Let , there a soft complex Cauchy sequence if for all SRNs , for exists a positive soft integer such that . all For to be less than the difference between any two soft elements that occur before and following the term’s . be a sequence of SCNs that conProposition 2 Let verges. Then it is bounded. converge to a Proof. Suppose that the SCSQ . Then for soft complex limit

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is bounded. be a bounded sequence of SCNs. Theorem 5 Let has a convergent subsequence. Then is bounded sequence of SCNs, then Proof. Since and are bounded sequences of SRNs. By and has a convergent subseLemma 1, and , respectively. Now is quences , and so a subsequence of the convergent sequence is converges to the same soft complex limit. Hence, as its real and imaginary a convergent subsequence of parts are both convergent sequence of SCNs. is a SCSQ bounded and converges to Theorem 6 If . Then . soft complex limit be a SCSQ bounded and it is conProof. Suppose . So, by Definition 18, for a SRN , there verge to , , is a soft natural integer in which is bounded, therefore, , this also since means that

inequality has given us this implies that . Now it gives (7)

implies that . This completes the proof. is a sequence of SCNs Theorem 7 If is a Cauchy sequence iff both sequences of SRNs then and are Cauchy sequence of SRNs. is a Cauchy sequence, then Proof. If we suppose a positive soft integer such that for every SRN . Then for , it gives that

Conversely, suppose that both and are Cauchy , since is a sequence of SRNs. Then for every SRN Cauchy sequence, there exists a positive soft integer such that for all , then . Similarly, since is Cauchy then there exists an positive soft integer such that for all . Then . If , then for , it gives

So is a Cauchy sequence of SCNs. Theorem 8 Every soft Complex Cauchy sequence has a bound. Proof. Suppose that is a soft Complex Cauchy , sequence. Then by Definition 18, for every SRN there exists a positive integer such that , for . So for , it gives . all Theorem 9 Every convergent SCSQ is a soft complex Cauchy sequence. Proof. Let be a SCSQ and convergent to . For , there exists a positive integer such that

Theorem 10 Every soft complex Cauchy sequence is convergent. Proof. Let be any soft complex Cauchy sequence, where and are Cauchy sequences of SRNs. Then and are convergent sequences of SRNs to and , respectively. Convergence of Series in SCNs Let be a sequence of SCNs, we make a new sequence that is specified by , where is called the sequence of partial sums of sequence of SCNs . The sequence is symbolized by called an infinite series of SCNs. If exists then the series is claimed as convergent and is its sum, i.e.

If a series of SCNs is not convergent, it is called divergent. Example 4 The soft complex series does not convergent. Because the SCSQ of its parial sums is periodic and therefore doses not have a limit.

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Theorem 11 If the infinite series of SCNs is . convergent then Proof. Let partial sum of the series of SCNs. Suppose that be the , this the series is convergent and be the sum , we get that , by implies that , taking limit on both sides, it gives that

Remark 2 Consider the infinite series of SCNs if then is called remainder of the infinite series of SCNs. . If is the sum of infinite series of SCNs then of terms with SCNs is Theorem 12 A series , there exists a positive convergent iff for every SRN , for all integer such that and p ≥ 0. is convergent, and let Proof. Suppose be the partial sum of the series of SCNs. Suppose that the series is convergent and , for every SRN be a sum of series, it means that there exists a positive integer such that , . for all be the remainder Let , of the infinite series of SCNs, we get that , for all , so that , for all . So that, or , for all and p ≥ 0. Conversely, we know that and if is its parand . From tial sum then, for all and , it means that . p ≥ 0 so that , a series Definition 21 Let , there exists a posconverges to , if for every SRN , for all . itive integer such that converges absoDefinition 22 A SCNs series converges. lutely if converges absolutely Proposition 3 If the series converges. then, be Proof. Let the series of partial sums in the context of its convergence. , there exists a positive integer Then for every SRN . If then, such that

Finally, it gives that is a Cauchy sequence in SCNs so by Theorem 9 it is a convergent sequence in SCNs and is convergent.

be a SCSQ and Theorem 13 Let . Then the soft complex series iff the soft real series and . Proof. First we write the sequence of partial sums of the series as , where and . This implies that iff . Now by Theorem 1 on convergent SCSQs we get that iff and . Therefore, . Conversely, since and are partial sums of the real soft series and , respectively, so the result is obtained directly.

Conclusion

The SCNs offer a useful tool for organizing and managing ambiguous and incomplete data. One of its benefits is that they may articulate imprecise values with both real and imaginary features, which allows for a more in-depth description of complex circumstances. Due to its flexibility, it can be applied to the analysis of systems with complex randomness, which is frequent in many real-world applications. In light of the aforementioned benefits and flexibility of SCNs, by using SCNs and SRSQ, the novel notion of a SCSQ has been presented, and its link with SCNs has been demonstrated by a convergent series analysis. Further, the soft complex limit of SCSQs has been introduced, which is an initial route for the convergence of SCSQs. Some famous soft convergence theorems are presented. Important theorems such as the Bolzano-Weierstrass theorem have also been explored. Soft complex boundedness of SCSQs is also well described. There are also some related soft complex theorems present. Cauchy Sequences in SCNs have also been reflected, as is their relationship with bounded sequences in SCNs. Furthermore, by using SCSQs with specific characteristics, we introduce the infinite series and its convergent in SCNs. The presented work can be used in order to characterize the various notions of differential and integral calculus in SSET environment. Moreover, the various theorems and ideas in fuzzy measure theory and fuzzy functional analysis can also be modified for SSET environment using the SCNs, and SCSQs. The suggested approach has enormous potential in a number of different fields for its future possibilities. Scholars could further explore ways to improve the suggested approaches to better manage situations in the actual world when specific and vague knowledge coexist. Additionally, there is an exciting opportunity for combining the suggested idea with modern innovations like machine learning and artificial intelligence, where the structure’s versatility may help create stronger and more effective algorithms for pattern recognition and data analysis.

Authorship Contributions

Sigma J Eng Nat Sci, Vol. 43, No. 5, pp. 1751−1759, October, 2025

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

Statement On The Use Of Artificial Intelligence

Artificial intelligence was not used in the preparation of the article.

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