Single, Double, Triple, Quadruple, Quintuple-valued IndetermSoft Set

1. Preliminaries

We collect the basic terminology and notation used in what follows. The definitions in this paper are assumed to be finite.

1.1. Soft Set and IndetermSoft Set

The definitions of the Soft Set and the IndetermSoft Set are provided below. Soft Set is a parameterized family of subsets where each parameter selects relevant elements from a universe for decision modeling [1,2,3].

Definition 1

(Soft Set). [1] Let U be a universal set and E a set of parameters. A soft set over U is defined as an ordered pair $(F,E)$, where F is a mapping from E to the power set $\mathcal{P}\left(U\right)$:

$$F:E\to \mathcal{P}\left(U\right).$$

For each parameter $e\in E$, $F\left(e\right)\subseteq U$ represents the set of e-approximate elements in U, with $(F,E)$ forming a parameterized family of subsets of U.

Example 1

(Soft Set — Apartment Search by Parameters). A renter filters apartments using simple binary conditions. Let the universe be the set of available listings

$$U=\{u_1,u_2,u_3,u_4,u_5,u_6,u_7,u_8\}.$$

Let the parameter set be

$$E=\{\mathrm{near}\_\mathrm{subway},\mathrm{pet}\_\mathrm{friendly},\mathrm{balcony},\mathrm{under}\$1500,\mathrm{new}\_\mathrm{build}\}.$$

Define a soft set $(F,E)$ with $F:E\to \mathcal{P}\left(U\right)$:

$$\begin{array}{cc}\hfill F(\mathrm{near}\_\mathrm{subway})& =\{u_1,u_2,u_5,u_6,u_8\},\hfill \\ \hfill F(\mathrm{pet}\_\mathrm{friendly})& =\{u_2,u_3,u_6,u_7\},\hfill \\ \hfill F\left(\mathrm{balcony}\right)& =\{u_1,u_3,u_4,u_6\},\hfill \\ \hfill F\left(\mathrm{under}\$1500\right)& =\{u_2,u_4,u_5,u_7\},\hfill \\ \hfill F(\mathrm{new}\_\mathrm{build})& =\{u_1,u_5,u_8\}.\hfill \end{array}$$

Each $F\left(e\right)\subseteq U$ collects listings relevant under parameter e.

Concrete queries with explicit computations.

$$\begin{array}{cc}\hfill \left(i\right)\mathit{near}\_\mathit{subway}\&\mathit{under}\$1500:& F(\mathrm{near}\_\mathrm{subway})\cap F\left(\mathrm{under}\$1500\right)=\{u_1,u_2,u_5,u_6,u_8\}\cap \{u_2,u_4,u_5,u_7\}\hfill \\ & =\{u_2,u_5\},|·|=2.\hfill \\ \hfill \left(\mathit{ii}\right)\mathit{pet}\_\mathit{friendly}\mathit{or}\mathit{balcony}:& F(\mathrm{pet}\_\mathrm{friendly})\cup F\left(\mathrm{balcony}\right)=\{u_2,u_3,u_6,u_7\}\cup \{u_1,u_3,u_4,u_6\}\hfill \\ & =\{u_1,u_2,u_3,u_4,u_6,u_7\},|·|=6.\hfill \\ \hfill \left(\mathit{iii}\right)\mathit{new}\_\mathit{build}\&\mathit{balcony}:& F(\mathrm{new}\_\mathrm{build})\cap F\left(\mathrm{balcony}\right)=\{u_1,u_5,u_8\}\cap \{u_1,u_3,u_4,u_6\}\hfill \\ & =\left\{u_1\right\},|·|=1.\hfill \end{array}$$

These set-theoretic combinations show how a soft set supports multi-constraint filtering with exact results.

Single-valued IndetermSoft Set maps each attribute value to one subset capturing undirected, non-unique indeterminacy over H; domain/codomain may be indeterminate [4,5,6,7,8].

Definition 2 ((Single-valued) IndetermSoft set)). [9,10,11,12] Let U be a universe of discourse, $H\subseteq U$ a non-empty subset, and $P\left(H\right)$ the powerset of H. Let A be the set of attribute values for an attribute a. A function $F:A\to P\left(H\right)$ is called an IndetermSoft Set if at least one of the following conditions holds:

• A has some indeterminacy.
• $P\left(H\right)$ has some indeterminacy.
• There exists at least one $v\in A$ such that $F\left(v\right)$ is indeterminate (unclear, uncertain, or not unique).
• Any two or all three of the above conditions.

An IndetermSoft Set is represented mathematically as:

$$F:A\to H(\cap ,\cup ,\oplus ,\neg ),$$

where $H(\cap ,\cup ,\oplus ,\neg )$ represents a structure closed under the IndetermSoft operators.

Example 2 ((Single-valued) IndetermSoft Set — Identity Verification Queue). Context. A hiring team triages applicants whose identity documents have quality issues. Let the review subset be

$$H=\{a_1,a_2,a_3,a_4,a_5,a_6,a_7\}\subseteq U,$$

and let the attribute be “document condition” with values

$$A=\{\mathrm{low}\_\mathrm{light}\_\mathrm{photo},\mathrm{partial}\_\mathrm{scan},\mathrm{metadata}\_\mathrm{mismatch}\}.$$

We define a single-valued IndetermSoft mapping $F:A\to \mathcal{P}\left(H\right)$, where $F\left(v\right)$ collects cases with undirected indeterminacy under v (neither truth- nor falsity-leaning).

Determinately specified slices.

$$\begin{array}{cc}\hfill F(\mathrm{low}\_\mathrm{light}\_\mathrm{photo})& =\{a_1,a_3,a_6\},\left|F\right|=3,\hfill \\ \hfill F(\mathrm{partial}\_\mathrm{scan})& =\{a_2,a_4\},\left|F\right|=2.\hfill \end{array}$$

Indeterminate slice (explicit non-uniqueness).  For $v^{★}=\mathit{metadata}\_\mathit{mismatch}$, conflicting HR and ATS logs produce two admissible subsets:

$$S_1=\left\{a_5\right\}(\mathit{only}\mathit{the}\mathit{strict}\mathit{mismatch}),S_2=\{a_5,a_7\}(\mathit{strict}+\mathit{borderline}\mathit{timestamp}\mathit{drift}).$$

Hence $F\left(v^{★}\right)$ is not uniquely determined:

$$F(\mathrm{metadata}\_\mathrm{mismatch})\in \left\{S_1,S_2\right\}=\left\{\left\{a_5\right\},\{a_5,a_7\}\right\}.$$

This satisfies condition (3) of the IndetermSoft definition (indeterminate image).

Let the determinate union be

$$L:=F(\mathrm{low}\_\mathrm{light}\_\mathrm{photo})\cup F(\mathrm{partial}\_\mathrm{scan})=\{a_1,a_2,a_3,a_4,a_6\}.$$

Including the indeterminate value yields an upper bound

$$U:=L\cup F(\mathrm{metadata}\_\mathrm{mismatch})\in \left\{L\cup S_1,L\cup S_2\right\}=\left\{\{a_1,a_2,a_3,a_4,a_5,a_6\},\{a_1,a_2,a_3,a_4,a_5,a_6,a_7\}\right\}.$$

Therefore the review queue size is bounded as

$$\left|L\right|=5\le \left|U\right|\in \{6,7\}.$$

This concrete non-uniqueness of $F\left(v^{★}\right)$ exemplifies a (single-valued) IndetermSoft set: one subset per value, with explicit indeterminacy captured by alternative admissible images.

1.2. Double-Valued Neutrosophic Set

Neutrosophic Set assigns to each element independent degrees of truth, indeterminacy, and falsity, enabling nuanced uncertainty modeling beyond fuzzy sets [13,14,15]. Double-Valued Neutrosophic Set records truth, truth-leaning indeterminacy, falsity-leaning indeterminacy, and falsity degrees for each element, capturing polarized ambiguous evidence effectively[16,17,18]. We consider Double-Valued Neutrosophic Logic, as outlined below (cf. [16,18,19]). Related concepts, such as the Multi-Valued Neutrosophic Set [20,21], are also well known.

Definition 3

(Double-Valued Neutrosophic Set). [16] Let X be a space of points (or objects) where each $x\in X$ represents an element. A Double-Valued Neutrosophic Set (DVNS) A is characterized by:

$$A=\{(x,T_A\left(x\right),I_T\left(x\right),I_F\left(x\right),F_A\left(x\right)):x\in X\},$$

where:

• $T_A\left(x\right)\in [0,1]$ is the truth membership value,
• $I_T\left(x\right)\in [0,1]$ is the indeterminacy leaning towards truth,
• $I_F\left(x\right)\in [0,1]$ is the indeterminacy leaning towards falsity,
• $F_A\left(x\right)\in [0,1]$ is the falsity membership value.

These values satisfy the condition:

$$0\le T_A\left(x\right)+I_T\left(x\right)+I_F\left(x\right)+F_A\left(x\right)\le 4.$$

Example 3

(Double-Valued Neutrosophic Set — Phishing Email Screening). Universe and target. Let X be the set of incoming emails and let $A\subseteq {[0,1]}^4$ encode a DVNS for the concept LegitimateMail. For each email $x\in X$ we assign

$$\left(T_A\left(x\right),I_T\left(x\right),I_F\left(x\right),F_A\left(x\right)\right),$$

where $T_A$ is truth (legitimate), $F_A$ is falsity (phishing), $I_T$ leans toward legitimacy, and $I_F$ leans toward phishing.

Signals.  We aggregate authentication (SPF/DKIM/DMARC), sender reputation, URL/attachment analysis, and writing-style features. Ambiguous signals (e.g., one pass and one fail) contribute to $I_T$ or $I_F$; consistent evidence contributes to $T_A$ or $F_A$.

Concrete assignments (three emails).

$$\begin{array}{ccccc}\mathrm{email}& T_A& I_T& I_F& F_A\\ e_1& 0.72& 0.14& 0.06& 0.08\\ e_2& 0.38& 0.20& 0.22& 0.20\\ e_3& 0.10& 0.08& 0.27& 0.55\end{array}$$

Justification. $e_1$: known domain, DMARC pass ⇒ large $T_A$; single shortened URL ⇒ small $I_F$. $e_2$: mixed signals (fresh domain, but no payload) ⇒ comparable $I_T$ and $I_F$. $e_3$: macro-enabled attachment and domain spoof patterns ⇒ large $F_A$.

Numerical verification.  Row sums:

$$\begin{array}{cc}\hfill e_1:& 0.72+0.14+0.06+0.08=1.00\le 4,\hfill \\ \hfill e_2:& 0.38+0.20+0.22+0.20=1.00\le 4,\hfill \\ \hfill e_3:& 0.10+0.08+0.27+0.55=1.00\le 4.\hfill \end{array}$$

Thus each quadruple satisfies the DVNS constraint $0\le T_A+I_T+I_F+F_A\le 4$.

1.3. Triple-Valued Neutrosophic Set (TVNS)

A Triple-Valued Neutrosophic Set assigns each element degrees for truth, truth-leaning indeterminacy, neutral indeterminacy, falsity-leaning indeterminacy, and falsity, within limits. In this subsection, we examine the concept of the Triple-Valued Neutrosophic Set (TVNS) [22,23,24].

Definition 4 (Triple-Valued Neutrosophic Set (TVNS)). [24] A Triple-Valued Neutrosophic Set A on X is defined as

$$A=\left\{\left(x,T_A\left(x\right),I_T\left(x\right),I_N\left(x\right),I_F\left(x\right),F_A\left(x\right)\right):x\in X\right\},$$

where

• $T_A\left(x\right)\in [0,1]$ is the truth membership degree,
• $I_T\left(x\right)\in [0,1]$ is the indeterminacy leaning towards truth,
• $I_N\left(x\right)\in [0,1]$ is the neutral indeterminacy (i.e., completely indeterminate, neither leaning towards truth nor falsity),
• $I_F\left(x\right)\in [0,1]$ is the indeterminacy leaning towards falsity,
• $F_A\left(x\right)\in [0,1]$ is the falsity membership degree.

For each $x\in X$, we have

$$0\le T_A\left(x\right)+I_T\left(x\right)+I_N\left(x\right)+I_F\left(x\right)+F_A\left(x\right)\le 5.$$

Example 4

(Triple-Valued Neutrosophic Set — Loan Approval Under Missing Evidence). Universe and target. Let X be the set of loan applications and consider the TVNS for the concept Creditworthy. For each applicant $x\in X$ we assign

$$\left(T_A\left(x\right),I_T\left(x\right),I_N\left(x\right),I_F\left(x\right),F_A\left(x\right)\right),$$

where $I_N$ is neutral indeterminacy due to missing or non-informative evidence.

Evidence channels.  Credit score and repayment history (push $T_A$), soft positives such as stable address (push $I_T$), missing employer verification or frozen bureau file (push $I_N$), soft negatives like recent inquiries (push $I_F$), and hard negatives such as charge-offs (push $F_A$).

Concrete assignments (three applicants).

$$\begin{array}{cccccc}\mathrm{applicant}& T_A& I_T& I_N& I_F& F_A\\ a_1& 0.55& 0.15& 0.10& 0.08& 0.12\\ a_2& 0.32& 0.18& 0.30& 0.10& 0.10\\ a_3& 0.12& 0.08& 0.20& 0.25& 0.35\end{array}$$

Justification. $a_1$: strong score and verified income ⇒ larger $T_A$; a minor document issue ⇒ small $I_N$. $a_2$: acceptable score but employer response pending ⇒ elevated $I_N$; mixed soft signals split between $I_T$ and $I_F$. $a_3$: prior delinquency and high utilization ⇒ large $F_A$; some uncollected data adds to $I_N$.

Numerical verification.  Row sums:

$$\begin{array}{cc}\hfill a_1:& 0.55+0.15+0.10+0.08+0.12=1.00\le 5,\hfill \\ \hfill a_2:& 0.32+0.18+0.30+0.10+0.10=1.00\le 5,\hfill \\ \hfill a_3:& 0.12+0.08+0.20+0.25+0.35=1.00\le 5.\hfill \end{array}$$

Hence each quintuple satisfies the TVNS constraint $0\le T_A+I_T+I_N+I_F+F_A\le 5$.

1.4. Quadruple-Valued Neutrosophic Set (QVNS)

A Quadruple-Valued Neutrosophic Set assigns each element truth, strongly truth-leaning indeterminacy, truth-leaning indeterminacy, falsity-leaning indeterminacy, strongly falsity-leaning indeterminacy, and falsity. The definition of the Quadruple-Valued Neutrosophic Set (QnVNS) is provided below [24,25].

Definition 5 (Quadruple-Valued Neutrosophic Set (QVNS)). [24] Let X be a universe of discourse. A Quadruple-Valued Neutrosophic Set A on X is defined as

$$A=\left\{\left(x,T_A\left(x\right),I_T^s\left(x\right),I_T\left(x\right),I_F\left(x\right),I_F^s\left(x\right),F_A\left(x\right)\right):x\in X\right\},$$

where:

• $T_A\left(x\right)\in [0,1]$ is the truth membership degree,
• $I_T^s\left(x\right)\in [0,1]$ is the indeterminacy strongly leaning towards truth,
• $I_T\left(x\right)\in [0,1]$ is the indeterminacy leaning towards truth,
• $I_F\left(x\right)\in [0,1]$ is the indeterminacy leaning towards falsity,
• $I_F^s\left(x\right)\in [0,1]$ is the indeterminacy strongly leaning towards falsity,
• $F_A\left(x\right)\in [0,1]$ is the falsity membership degree.

For each $x\in X$, it holds that

$$0\le T_A\left(x\right)+I_T^s\left(x\right)+I_T\left(x\right)+I_F\left(x\right)+I_F^s\left(x\right)+F_A\left(x\right)\le 6.$$

Example 5

(Quadruple-Valued Neutrosophic Set — E-commerce Product Authenticity). Universe and target. Let X be items listed on a marketplace. Consider the QVNS for the concept AuthenticItem . For each item $x\in X$ we assign

$$\left(T_A\left(x\right),I_T^s\left(x\right),I_T\left(x\right),I_F\left(x\right),I_F^s\left(x\right),F_A\left(x\right)\right),$$

where $T_A$/$F_A$ are truth/falsity memberships (authentic vs. counterfeit), and $I_T^s,I_T,I_F,I_F^s$ are indeterminacy slices (strong/ordinary truth- or falsity-leaning).

Evidence channels.  Manufacturer serial check (pushes $I_T^s$ or $T_A$ when strongly positive), invoice validation (pushes $I_T$), packaging/label anomalies (push $I_F$), known counterfeit patterns (push $I_F^s$), and third-party lab confirmation (boosts $T_A$ or $F_A$).

Concrete assignments (three items).

$$\begin{array}{ccccccc}\mathrm{item}& T_A& I_T^s& I_T& I_F& I_F^s& F_A\\ i_1& 0.60& 0.15& 0.10& 0.05& 0.03& 0.07\\ i_2& 0.30& 0.05& 0.20& 0.25& 0.10& 0.10\\ i_3& 0.08& 0.02& 0.10& 0.25& 0.35& 0.20\end{array}$$

Justification. $i_1$: serial verified and lab spot-check ⇒ large $T_A$, some $I_T^s$; a minor label scuff ⇒ small $I_F$. $i_2$: mixed invoice and seller history ⇒ comparable $I_T$ and $I_F$; moderate $T_A$. $i_3$: hallmark counterfeit indicators ⇒ large $I_F^s$ and $F_A$.

Numerical verification.  For each row,

$$\begin{array}{cc}\hfill i_1:& 0.60+0.15+0.10+0.05+0.03+0.07=1.00\le 6,\hfill \\ \hfill i_2:& 0.30+0.05+0.20+0.25+0.10+0.10=1.00\le 6,\hfill \\ \hfill i_3:& 0.08+0.02+0.10+0.25+0.35+0.20=1.00\le 6.\hfill \end{array}$$

Thus each sextuple satisfies $0\le T_A+I_T^s+I_T+I_F+I_F^s+F_A\le 6$.

1.5. Quintuple-Valued Neutrosophic Set (QnVNS)

A Quintuple-Valued Neutrosophic Set assigns each element truth; strong truth-leaning, truth-leaning, neutral, falsity-leaning, strong falsity-leaning indeterminacies; and falsity, with bounds. The definition of the Quintuple-Valued Neutrosophic Set (QnVNS) is provided below. [24]

Definition 6 (Quintuple-Valued Neutrosophic Set (QnVNS)). [24] Let X be a universe of discourse. A Quintuple-Valued Neutrosophic Set A on X is defined as

$$A=\left\{\left(x,T_A\left(x\right),I_T^s\left(x\right),I_T\left(x\right),I_N\left(x\right),I_F\left(x\right),I_F^s\left(x\right),F_A\left(x\right)\right):x\in X\right\},$$

where:

• $T_A\left(x\right)\in [0,1]$ is the truth membership degree,
• $I_T^s\left(x\right)\in [0,1]$ is the indeterminacy strongly leaning towards truth,
• $I_T\left(x\right)\in [0,1]$ is the indeterminacy leaning towards truth,
• $I_N\left(x\right)\in [0,1]$ represents neutral indeterminacy(i.e., completely indeterminate, neither leaning towards truth nor falsity),
• $I_F\left(x\right)\in [0,1]$ is the indeterminacy leaning towards falsity,
• $I_F^s\left(x\right)\in [0,1]$ is the indeterminacy strongly leaning towards falsity,
• $F_A\left(x\right)\in [0,1]$ is the falsity membership degree.

For each $x\in X$, it holds that

$$0\le T_A\left(x\right)+I_T^s\left(x\right)+I_T\left(x\right)+I_N\left(x\right)+I_F\left(x\right)+I_F^s\left(x\right)+F_A\left(x\right)\le 7.$$

Example 6

(Quintuple-Valued Neutrosophic Set — Product Recall Impact Assessment). Universe and target. Let X be purchased product units. Consider the QnVNS for the concept AffectedByRecall. For each unit $x\in X$ we assign

$$\left(T_A\left(x\right),I_T^s\left(x\right),I_T\left(x\right),I_N\left(x\right),I_F\left(x\right),I_F^s\left(x\right),F_A\left(x\right)\right),$$

where $I_N$ is neutral indeterminacy (e.g., missing batch code).

Evidence channels.  Lot/batch match (push $I_T^s$/$T_A$), weak cues such as packaging date proximity (push $I_T$), missing or unreadable label (push $I_N$), retailer confirmation of safe lot (push $I_F$), and manufacturer certificate of unaffected series (push $I_F^s$/$F_A$).

Concrete assignments (three units).

$$\begin{array}{cccccccc}\mathrm{unit}& T_A& I_T^s& I_T& I_N& I_F& I_F^s& F_A\\ b_1& 0.55& 0.20& 0.10& 0.05& 0.05& 0.02& 0.03\\ b_2& 0.20& 0.05& 0.15& 0.30& 0.15& 0.05& 0.10\\ b_3& 0.08& 0.02& 0.10& 0.10& 0.20& 0.25& 0.25\end{array}$$

Justification. $b_1$: exact lot match and vendor alert ⇒ large $T_A$ and $I_T^s$. $b_2$: label unreadable ⇒ elevated $I_N$; weak context on both sides. $b_3$: certificate proves different series ⇒ higher $I_F^s$ and $F_A$.

Numerical verification.  For each row,

$$\begin{array}{cc}\hfill b_1:& 0.55+0.20+0.10+0.05+0.05+0.02+0.03=1.00\le 7,\hfill \\ \hfill b_2:& 0.20+0.05+0.15+0.30+0.15+0.05+0.10=1.00\le 7,\hfill \\ \hfill b_3:& 0.08+0.02+0.10+0.10+0.20+0.25+0.25=1.00\le 7.\hfill \end{array}$$

Hence each septuple satisfies $0\le T_A+I_T^s+I_T+I_N+I_F+I_F^s+F_A\le 7$.

1.6. Double-Valued Indetermsoft Set

Double-valued Indetermsoft Set maps each attribute value to two subsets: truth-leaning indeterminacy and falsity-leaning indeterminacy within a universe, allowing overlaps. The definition of the Double-valued Indetermsoft Set is provided below. [24]

Definition 7

(Double-valued Indetermsoft Set). [24] Let U be a universe of discourse, $H\subseteq U$ a non-empty subset, and $\mathcal{P}\left(H\right)$ the power set of H. Let A be a set of attribute values associated with a specific attribute a. A function

$$F:A\to \mathcal{P}\left(H\right)\times \mathcal{P}\left(H\right)$$

is called a Double-valued Indetermsoft Set if any of the following conditions hold:

• The set A exhibits some degree of indeterminacy.
• The power set $\mathcal{P}\left(H\right)$ exhibits some level of indeterminacy.
• There exists at least one attribute value $v\in A$ such that the ordered pair

  $$F\left(v\right)=\left(F_T\left(v\right),F_F\left(v\right)\right)$$
  is indeterminate, in the sense that either $F_T\left(v\right)$ or $F_F\left(v\right)$ (or both) is unclear, uncertain, or not uniquely defined.
• Any combination of the above conditions holds.

Here, for each $v\in A$,

• $F_T\left(v\right)\subseteq H$ represents the indeterminacy leaning towards truth associated with v, and
• $F_F\left(v\right)\subseteq H$ represents the indeterminacy leaning towards falsity associated with v.

Example 7

(Double-valued Indetermsoft Set — Warranty Claims Triage). Let the universe H be a finite set of consumer-electronics warranty claims

$$H=\{c_1,c_2,c_3,c_4,c_5,c_6,c_7\}.$$

Fix an attribute “document reliability,” with attribute values

$$A=\{\mathrm{blurred}\_\mathrm{receipt},\mathrm{third}\_\mathrm{party}\_\mathrm{repair},\mathrm{serial}\_\mathrm{OCR}\_\mathrm{error}\}.$$

Define a Double-valued Indetermsoft Set $F:A\to \mathcal{P}\left(H\right)\times \mathcal{P}\left(H\right)$ by

$$\begin{array}{cc}\hfill F(\mathrm{blurred}\_\mathrm{receipt})& =\left(F_T,F_F\right)=\left(\{c_1,c_2,c_5\},\left\{c_3\right\}\right),\hfill \\ \hfill F(\mathrm{third}\_\mathrm{party}\_\mathrm{repair})& =\left(\left\{c_5\right\},\{c_4,c_6\}\right),\hfill \\ \hfill F(\mathrm{serial}\_\mathrm{OCR}\_\mathrm{error})& =\left(\{c_2,c_6\},\left\{c_7\right\}\right).\hfill \end{array}$$

Interpretation.

• $F_T\left(v\right)$ (truth-leaning indeterminacy): claims whose evidence likely supports validity under v, but is still indeterminate (e.g., other documents corroborate a blurred receipt).
• $F_F\left(v\right)$ (falsity-leaning indeterminacy): claims whose evidence likely undermines validity under v (e.g., non-authorized third-party repair), yet not conclusively.

Concrete checks (cardinalities and overlaps across different v):

$$\begin{array}{cc}& |F_T(\mathrm{blurred}\_\mathrm{receipt})|=3,|F_F(\mathrm{blurred}\_\mathrm{receipt})|=1,\left|F_T\cup F_F\right|=4,\hfill \\ & |F_T(\mathrm{third}\_\mathrm{party}\_\mathrm{repair})|=1,|F_F(\mathrm{third}\_\mathrm{party}\_\mathrm{repair})|=2,\left|F_T\cup F_F\right|=3,\hfill \\ & |F_T(\mathrm{serial}\_\mathrm{OCR}\_\mathrm{error})|=2,|F_F(\mathrm{serial}\_\mathrm{OCR}\_\mathrm{error})|=1,\left|F_T\cup F_F\right|=3.\hfill \end{array}$$

Cross-attribute tension is visible: for instance, $c_6\in F_F(\mathrm{third}\_\mathrm{party}\_\mathrm{repair})$ but $c_6\in F_T(\mathrm{serial}\_\mathrm{OCR}\_\mathrm{error})$, capturing conflicting indeterminate signals that motivate a second-stage manual review.

2. Main Results

In this section, we present the main contributions of this paper, focusing on the concepts of ContraSoft Set and ContraRough Set.

2.1. Triple-Valued Indetermsoft Set

Triple-valued Indetermsoft Set maps each attribute value to three subsets: truth-leaning, neutral, and falsity-leaning indeterminacy over a universe, allowing overlaps.

Definition 8 (Triple-valued Indetermsoft Set (TV-ISS)). Let U be a universe, $H\subseteq U$ nonempty, and A a set of attribute values. A TV-ISS is a mapping

$$G:A⟶\mathcal{P}{\left(H\right)}^3,G\left(v\right)=\left(G_T\left(v\right),G_N\left(v\right),G_F\left(v\right)\right),$$

with the following intended semantics for each $v\in A$:

• $G_T\left(v\right)\subseteq H$: indeterminacy leaning to truth,
• $G_N\left(v\right)\subseteq H$: neutral indeterminacy (neither leaning to truth nor to falsity),
• $G_F\left(v\right)\subseteq H$: indeterminacy leaning to falsity.

No exclusivity is imposed a priori; intersections are allowed if the application dictates. When needed, one may assume $G_T\left(v\right),G_N\left(v\right),G_F\left(v\right)$ are pairwise disjoint to model a partition of the indeterminate region.

Example 8

(Concrete TV-ISS, its projection, and the embedding). Let the finite universe and attribute-values be

$$H=\{h_1,h_2,h_3,h_4\},A=\{\mathrm{easy},\mathrm{hard}\}.$$

Define a TV-ISS $G:A\to \mathcal{P}{\left(H\right)}^3$ by

$$\begin{array}{cc}\hfill G\left(\mathrm{easy}\right)& =\left(\{h_1,h_2\},\left\{h_3\right\},\left\{h_4\right\}\right),\hfill \\ \hfill G\left(\mathrm{hard}\right)& =\left(\left\{h_1\right\},\{h_2,h_3\},\left\{h_4\right\}\right).\hfill \end{array}$$

Cardinalities are

$$\begin{array}{cc}& |G_T\left(\mathrm{easy}\right)|=2,|G_N\left(\mathrm{easy}\right)|=1,|G_F\left(\mathrm{easy}\right)|=1,\hfill \\ & |G_T\left(\mathrm{hard}\right)|=1,|G_N\left(\mathrm{hard}\right)|=2,|G_F\left(\mathrm{hard}\right)|=1.\hfill \end{array}$$

The canonical DV-ISS projection $\Pi \left(G\right)$ and upper envelope $\overline{\Pi }\left(G\right)$ are

$$\begin{array}{cccc}\hfill \Pi \left(G\right)\left(\mathrm{easy}\right)& =\left(\{h_1,h_2\},\left\{h_4\right\}\right),\hfill & \hfill \overline{\Pi }\left(G\right)\left(\mathrm{easy}\right)& =\left(\{h_1,h_2,h_3\},\{h_3,h_4\}\right),\hfill \\ \hfill \Pi \left(G\right)\left(\mathrm{hard}\right)& =\left(\left\{h_1\right\},\left\{h_4\right\}\right),\hfill & \hfill \overline{\Pi }\left(G\right)\left(\mathrm{hard}\right)& =\left(\{h_1,h_2,h_3\},\{h_2,h_3,h_4\}\right),\hfill \end{array}$$

and one checks the coordinatewise inclusions

$$\Pi \left(G\right)⪯\overline{\Pi }\left(G\right),$$

since, for instance, $\{h_1,h_2\}\subseteq \{h_1,h_2,h_3\}$ and $\left\{h_4\right\}\subseteq \{h_3,h_4\}$ at $v=\mathrm{easy}$.

Now apply the DV→TV embedding ι to the DV-ISS $F:=\Pi \left(G\right)$:

$$\left(\iota \left(F\right)\right)\left(\mathrm{easy}\right)=\left(\{h_1,h_2\},⌀,\left\{h_4\right\}\right),\left(\iota \left(F\right)\right)\left(\mathrm{hard}\right)=\left(\left\{h_1\right\},⌀,\left\{h_4\right\}\right).$$

By construction,

$$(\Pi \circ \iota )\left(F\right)=F$$

holds pointwise and componentwise, e.g.,

$$(\Pi \circ \iota )\left(F\right)\left(\mathrm{easy}\right)=\Pi \left(\{h_1,h_2\},⌀,\left\{h_4\right\}\right)=\left(\{h_1,h_2\},\left\{h_4\right\}\right)=F\left(\mathrm{easy}\right).$$

This verifies, with explicit sets and equalities, the identity $\Pi \circ \iota =\mathrm{id}$ from Theorem 1.

Example 9

(Triple-valued Indetermsoft Set — Hiring Degree Verification). Let the universe H be a finite set of applicants

$$H=\{a_1,a_2,a_3,a_4,a_5,a_6\}.$$

Consider the attribute “credential verification,” with attribute values

$$A=\{\mathrm{degree}\_\mathrm{email}\_\mathrm{unreplied},\mathrm{transcript}\_\mathrm{low}\_\mathrm{quality}\}.$$

Define a Triple-valued Indetermsoft Set $G:A\to \mathcal{P}{\left(H\right)}^3$ by

$$\begin{array}{cc}\hfill G(\mathrm{degree}\_\mathrm{email}\_\mathrm{unreplied})& =\left(G_T,G_N,G_F\right)=\left(\{a_1,a_3\},\{a_2,a_5\},\left\{a_6\right\}\right),\hfill \\ \hfill G(\mathrm{transcript}\_\mathrm{low}\_\mathrm{quality})& =\left(\{a_3,a_4\},\left\{a_5\right\},\left\{a_2\right\}\right).\hfill \end{array}$$

Semantics for each $v\in A$:

• $G_T\left(v\right)$: truth-leaning indeterminacy (signals suggest the claim is likely true but not definitive yet; e.g., alumni registry hints).
• $G_N\left(v\right)$: neutral indeterminacy (insufficient information; e.g., awaiting registrar reply).
• $G_F\left(v\right)$: falsity-leaning indeterminacy (signals suggest likely false; e.g., domain mismatch, inconsistent graduation dates).

Concrete verifications (sizes and disjointness here by design):

$$\begin{array}{cc}& |G_T(\mathrm{degree}\_\mathrm{email}\_\mathrm{unreplied})|=2,|G_N|=2,|G_F|=1,\left|G_T\cup G_N\cup G_F\right|=5,\hfill \\ & |G_T(\mathrm{transcript}\_\mathrm{low}\_\mathrm{quality})|=2,|G_N|=1,|G_F|=1,\left|G_T\cup G_N\cup G_F\right|=4.\hfill \end{array}$$

If one wishes to compare with a Double-valued view, the canonical projection that discards the neutral slice is

$$F^{\downarrow }\left(v\right):=\left(G_T\left(v\right),G_F\left(v\right)\right),$$

so, for example, $F^{\downarrow }(\mathrm{degree}\_\mathrm{email}\_\mathrm{unreplied})=(\{a_1,a_3\},\left\{a_6\right\})$, which is ready for two-queue downstream handling (provisional accept vs. provisional reject) while cases in $G_N$ await additional evidence.

Theorem 1

(TV-ISS generalizes DV-ISS). There exist mappings

$$\iota :{\mathrm{ISS}}_2(A,H)\to {\mathrm{ISS}}_3(A,H),\Pi :{\mathrm{ISS}}_3(A,H)\to {\mathrm{ISS}}_2(A,H)$$

defined pointwise by

$$\left(\iota \left(F\right)\right)\left(v\right):=\left(F_T\left(v\right),⌀,F_F\left(v\right)\right),\left(\Pi \left(G\right)\right)\left(v\right):=\left(G_T\left(v\right),G_F\left(v\right)\right),$$

such that

$$\Pi \circ \iota ={\mathrm{id}}_{{\mathrm{ISS}}_2(A,H)}.$$

Consequently, ${\mathrm{ISS}}_2(A,H)$ is isomorphic to the sub-class

$${\mathrm{ISS}}_3^{\left(0\right)}(A,H):=\left\{G\in {\mathrm{ISS}}_3(A,H):G_N\left(v\right)=⌀\mathrm{for}\mathrm{all}v\in A\right\}\subseteq {\mathrm{ISS}}_3(A,H),$$

and every DV-ISS is a special case of a TV-ISS (obtained by taking all neutral slices empty).

Proof. 

Fix $F\in {\mathrm{ISS}}_2(A,H)$. By definition,

$$F:A\to \mathcal{P}\left(H\right)\times \mathcal{P}\left(H\right),v\mapsto \left(F_T\left(v\right),F_F\left(v\right)\right).$$

Define $\iota \left(F\right):A\to \mathcal{P}{\left(H\right)}^3$ by $\left(\iota \left(F\right)\right)\left(v\right)=\left(F_T\left(v\right),⌀,F_F\left(v\right)\right)$. This lies in ${\mathrm{ISS}}_3(A,H)$ since $⌀\in \mathcal{P}\left(H\right)$.

Next, for $G\in {\mathrm{ISS}}_3(A,H)$ with $G\left(v\right)=(G_T\left(v\right),G_N\left(v\right),G_F\left(v\right))$, define $\Pi \left(G\right)\left(v\right)=(G_T\left(v\right),G_F\left(v\right))$. Then $\Pi \left(G\right)\in {\mathrm{ISS}}_2(A,H)$.

We compute the composition explicitly, for each $v\in A$:

$$\left((\Pi \circ \iota )\left(F\right)\right)\left(v\right)=\Pi \left(\iota \left(F\right)\right)\left(v\right)=\Pi \left(F_T\left(v\right),⌀,F_F\left(v\right)\right)=\left(F_T\left(v\right),F_F\left(v\right)\right)=F\left(v\right).$$

Hence $\Pi \circ \iota ={\mathrm{id}}_{{\mathrm{ISS}}_2(A,H)}$.

Injectivity of $\iota$: if $\iota \left(F\right)=\iota \left(F^{\prime }\right)$, then for every v,

$$\left(F_T\left(v\right),⌀,F_F\left(v\right)\right)=\left(F_T^{\prime }\left(v\right),⌀,F_F^{\prime }\left(v\right)\right),$$

whence $F_T\left(v\right)=F_T^{\prime }\left(v\right)$ and $F_F\left(v\right)=F_F^{\prime }\left(v\right)$, so $F=F^{\prime }$.

Moreover, the image of $\iota$ is precisely ${\mathrm{ISS}}_3^{\left(0\right)}(A,H)$. Indeed, for any $G\in {\mathrm{ISS}}_3^{\left(0\right)}(A,H)$ one has $G_N\left(v\right)=⌀$ for all v, and setting $F\left(v\right)=(G_T\left(v\right),G_F\left(v\right))$ yields $G=\iota \left(F\right)$.

Therefore DV-ISS is canonically identified with the neutral-empty sub-class of TV-ISS, so TV-ISS strictly generalizes DV-ISS. □

2.2. Quadruple-Valued Indetermsoft Set

Quadruple-valued Indetermsoft Set assigns four indeterminacy slices per attribute value, mirroring Quadruple-valued Neutrosophic:

$$(\mathrm{strong}\mathrm{truth}\text{-}\mathrm{leaning}),\mathrm{truth}\text{-}\mathrm{leaning},\mathrm{falsity}\text{-}\mathrm{leaning},\mathrm{strong}\mathrm{falsity}\text{-}\mathrm{leaning}.$$

Definition 9 (Quadruple-valued Indetermsoft Set (QV-ISS)). Let U be a universe, $H\subseteq U$ nonempty, and A a set of attribute values. A QV-ISS is a mapping

$$Q:A⟶\mathcal{P}{\left(H\right)}^4,Q\left(v\right)=\left(Q_T^s\left(v\right),Q_T\left(v\right),Q_F\left(v\right),Q_F^s\left(v\right)\right),$$

where, for each $v\in A$:

• $Q_T^s\left(v\right)\subseteq H$:  strong truth-leaning indeterminacy  (very strong evidence toward truth, yet not fully resolved),
• $Q_T\left(v\right)\subseteq H$:  truth-leaning indeterminacy  (moderate evidence toward truth, still inconclusive),
• $Q_F\left(v\right)\subseteq H$:  falsity-leaning indeterminacy  (moderate evidence toward falsity, still inconclusive),
• $Q_F^s\left(v\right)\subseteq H$:  strong falsity-leaning indeterminacy  (very strong evidence toward falsity, yet not fully resolved).

Theorem 2

(QV-ISS generalizes TV-ISS). Define maps

$$\iota :{\left(\mathcal{P}{\left(H\right)}^3\right)}^A⟶{\left(\mathcal{P}{\left(H\right)}^4\right)}^A,\left(\iota \left(G\right)\right)\left(v\right):=\left(⌀,G_T\left(v\right)\cup G_N\left(v\right),G_F\left(v\right)\cup G_N\left(v\right),⌀\right),$$

and

$$\Pi :{\left(\mathcal{P}{\left(H\right)}^4\right)}^A⟶{\left(\mathcal{P}{\left(H\right)}^3\right)}^A,\left(\Pi \left(Q\right)\right)\left(v\right):=\left(T\left(v\right),N\left(v\right),F\left(v\right)\right),$$

where, writing $S_T\left(v\right):=Q_T^s\left(v\right)\cup Q_T\left(v\right)$ and $S_F\left(v\right):=Q_F^s\left(v\right)\cup Q_F\left(v\right)$,

$$T\left(v\right):=S_T\left(v\right)\setminus S_F\left(v\right),N\left(v\right):=S_T\left(v\right)\cap S_F\left(v\right),F\left(v\right):=S_F\left(v\right)\setminus S_T\left(v\right).$$

Then for every TV-ISS G in partition form, one has

$$(\Pi \circ \iota )\left(G\right)=G.$$

Consequently, TV-ISS embeds injectively into QV-ISS; hence QV-ISS (canonically) generalizes TV-ISS.

Proof. 

Fix G and $v\in A$. By definition of $\iota$,

$$Q_T^s=⌀,Q_T=G_T\cup G_N,Q_F=G_F\cup G_N,Q_F^s=⌀.$$

Hence

$$S_T=Q_T^s\cup Q_T=G_T\cup G_N,S_F=Q_F^s\cup Q_F=G_F\cup G_N.$$

Using the pairwise disjointness $G_T\cap G_N=G_N\cap G_F=G_T\cap G_F=⌀$, we compute:

$$\begin{array}{cc}\hfill T& =S_T\setminus S_F=(G_T\cup G_N)\setminus (G_F\cup G_N)=\left(G_T\setminus (G_F\cup G_N)\right)\cup \underset{=⌀}{\underbrace{(G_N\setminus (G_F\cup G_N))}}=G_T,\hfill \\ \hfill N& =S_T\cap S_F=(G_T\cup G_N)\cap (G_F\cup G_N)=\underset{=⌀}{\underbrace{(G_T\cap G_F)}}\cup \underset{=⌀}{\underbrace{(G_T\cap G_N)}}\cup \underset{=⌀}{\underbrace{(G_N\cap G_F)}}\cup G_N=G_N,\hfill \\ \hfill F& =S_F\setminus S_T=(G_F\cup G_N)\setminus (G_T\cup G_N)=\left(G_F\setminus (G_T\cup G_N)\right)\cup \underset{=⌀}{\underbrace{(G_N\setminus (G_T\cup G_N))}}=G_F.\hfill \end{array}$$

Therefore $(\Pi \circ \iota )\left(G\right)\left(v\right)=(G_T\left(v\right),G_N\left(v\right),G_F\left(v\right))=G\left(v\right)$ for every $v\in A$, i.e., $(\Pi \circ \iota )\left(G\right)=G$. Since $\Pi \circ \iota =\mathrm{id}$ on TV-ISS, $\iota$ is injective, providing an embedding of TV-ISS into QV-ISS. □

2.3. Quintuple-Valued Indetermsoft Set

Quintuple-valued Indetermsoft Set assigns five indeterminacy slices per value, mirroring Quintuple-valued Neutrosophic:

$$(\mathrm{strong}\mathrm{truth}\text{-}\mathrm{leaning}),\mathrm{truth}\text{-}\mathrm{leaning},\mathrm{neutral},\mathrm{falsity}\text{-}\mathrm{leaning},(\mathrm{strong}\mathrm{falsity}\text{-}\mathrm{leaning}).$$

Definition 10 (Quintuple-valued Indetermsoft Set (5V-ISS)). A 5V-ISS is a mapping

$$R:A\to \mathcal{P}{\left(H\right)}^5,R\left(v\right)=\left(R_T^s\left(v\right),R_T\left(v\right),R_N\left(v\right),R_F\left(v\right),R_F^s\left(v\right)\right),$$

with semantics, for each $v\in A$:

• $R_T^s\left(v\right)\subseteq H$:  strong truth-leaning indeterminacy,
• $R_T\left(v\right)\subseteq H$:  truth-leaning indeterminacy,
• $R_N\left(v\right)\subseteq H$:  neutral indeterminacy  (neither leaning toward truth nor falsity),
• $R_F\left(v\right)\subseteq H$:  falsity-leaning indeterminacy,
• $R_F^s\left(v\right)\subseteq H$: strong falsity-leaning indeterminacy.

Theorem 3

(5V-ISS generalizes QV-ISS). Define the maps

$$\iota :{\mathrm{ISS}}_4(A,H)\to {\mathrm{ISS}}_5(A,H),\left(\iota \left(Q\right)\right)\left(v\right):=\left(Q_T^s\left(v\right),Q_T\left(v\right),⌀,Q_F\left(v\right),Q_F^s\left(v\right)\right),$$

$$\Pi :{\mathrm{ISS}}_5(A,H)\to {\mathrm{ISS}}_4(A,H),\left(\Pi \left(R\right)\right)\left(v\right):=\left(R_T^s\left(v\right),R_T\left(v\right),R_F\left(v\right),R_F^s\left(v\right)\right).$$

Then

$$\Pi \circ \iota ={\mathrm{id}}_{{\mathrm{ISS}}_4(A,H)}.$$

Consequently, ${\mathrm{ISS}}_4(A,H)$ is canonically isomorphic to the subclass

$${\mathrm{ISS}}_5^{\left(0\right)}(A,H):=\left\{R\in {\mathrm{ISS}}_5(A,H):R_N\left(v\right)=⌀\mathrm{for}\mathrm{all}v\in A\right\}\subset {\mathrm{ISS}}_5(A,H),$$

hence 5V-ISS strictly extends QV-ISS whenever some $R_N\left(v\right)\ne ⌀$.

Proof. 

Let $Q\in {\mathrm{ISS}}_4(A,H)$ and $v\in A$. By definition of $\iota$ and $\Pi$,

$$(\Pi \circ \iota )\left(Q\right)\left(v\right)=\Pi \left(Q_T^s\left(v\right),Q_T\left(v\right),⌀,Q_F\left(v\right),Q_F^s\left(v\right)\right)=\left(Q_T^s\left(v\right),Q_T\left(v\right),Q_F\left(v\right),Q_F^s\left(v\right)\right)=Q\left(v\right).$$

Thus $\Pi \circ \iota =\mathrm{id}$ pointwise and componentwise. Injectivity of $\iota$: if $\iota \left(Q\right)=\iota \left(Q^{\prime }\right)$, then for every v

$$(Q_T^s,Q_T,⌀,Q_F,Q_F^s)=(Q_T^{s\prime },Q_T^{\prime },⌀,Q_F^{\prime },Q_F^{s\prime }),$$

which forces $Q=Q^{\prime }$. The image of $\iota$ is exactly ${\mathrm{ISS}}_5^{\left(0\right)}(A,H)$ because $R\in {\mathrm{ISS}}_5(A,H)$ belongs to ${\mathrm{ISS}}_5^{\left(0\right)}(A,H)$ iff $R_N\equiv ⌀$, in which case $R=\iota \left(Q\right)$ with $Q\left(v\right)=(R_T^s\left(v\right),R_T\left(v\right),R_F\left(v\right),R_F^s\left(v\right))$. To see the extension is strict, fix any $h^{★}\in H$ and $v^{★}\in A$ and define $R\in {\mathrm{ISS}}_5(A,H)$ by $R_N\left(v^{★}\right)=\left\{h^{★}\right\}$ and the remaining coordinates as in $\iota \left(Q\right)$ for some Q; then $R\notin {\mathrm{ISS}}_5^{\left(0\right)}(A,H)$, so R is not in the image of $\iota$. □

Example 10

(Concrete embedding and equality check with counts). Let

$$H=\{h_1,h_2,h_3,h_4,h_5\},A=\{\mathrm{easy},\mathrm{hard}\}.$$

Define a QV-ISS $Q\in {\mathrm{ISS}}_4(A,H)$ by

$$\begin{array}{cc}\hfill Q\left(\mathrm{easy}\right)& =\left(\left\{h_1\right\},\left\{h_2\right\},\left\{h_4\right\},\left\{h_5\right\}\right),\hfill \\ \hfill Q\left(\mathrm{hard}\right)& =\left(\{h_1,h_3\},\left\{h_2\right\},\left\{h_4\right\},\left\{h_5\right\}\right).\hfill \end{array}$$

Cardinalities:

$$\begin{array}{cc}& |Q_T^s\left(\mathrm{easy}\right)|=1,|Q_T\left(\mathrm{easy}\right)|=1,|Q_F\left(\mathrm{easy}\right)|=1,|Q_F^s\left(\mathrm{easy}\right)|=1;\hfill \\ & |Q_T^s\left(\mathrm{hard}\right)|=2,|Q_T\left(\mathrm{hard}\right)|=1,|Q_F\left(\mathrm{hard}\right)|=1,|Q_F^s\left(\mathrm{hard}\right)|=1.\hfill \end{array}$$

Embed via ι to obtain a 5V-ISS $R:=\iota \left(Q\right)$:

$$\begin{array}{cc}\hfill R\left(\mathrm{easy}\right)& =\left(\left\{h_1\right\},\left\{h_2\right\},⌀,\left\{h_4\right\},\left\{h_5\right\}\right),\hfill \\ \hfill R\left(\mathrm{hard}\right)& =\left(\{h_1,h_3\},\left\{h_2\right\},⌀,\left\{h_4\right\},\left\{h_5\right\}\right).\hfill \end{array}$$

Project back:

$$\Pi \left(R\right)\left(\mathrm{easy}\right)=\left(\left\{h_1\right\},\left\{h_2\right\},\left\{h_4\right\},\left\{h_5\right\}\right)=Q\left(\mathrm{easy}\right),\Pi \left(R\right)\left(\mathrm{hard}\right)=\left(\{h_1,h_3\},\left\{h_2\right\},\left\{h_4\right\},\left\{h_5\right\}\right)=Q\left(\mathrm{hard}\right).$$

Hence $(\Pi \circ \iota )\left(Q\right)=Q$ withliteralset equality in all coordinates. To witness strict extension, define $R^{\prime }$ by taking $R_N^{\prime }\left(\mathrm{easy}\right)=\left\{h_3\right\}$ and all other coordinates equal to R; then $R^{\prime }\notin {\mathrm{ISS}}_5^{\left(0\right)}(A,H)$, so there is no $Q^{\prime }$ with $R^{\prime }=\iota \left(Q^{\prime }\right)$.

3. Conclusions

In this paper, we defined five variants: (i) Single-valued, mapping each attribute value to one subset capturing undirected indeterminacy; (ii) Double-valued, splitting into truth-leaning and falsity-leaning slices; (iii) Triple-valued, adding a neutral slice; (iv) Quadruple-valued, refining polarity by distinguishing strong and ordinary truth/falsity leanings; and (v) Quintuple-valued, further introducing a central neutral band between the two strengths. For clarity, the overall summary of the IndetermSoft sets discussed in this paper is presented in Table 1.

In the future, we intend to extend the concepts developed in this paper by employing HyperSoft Sets [26,27,28], SuperHyperSoft Sets [29,30,31], TreeSoft Sets [32,33], ForestSoft Sets [34,35,36], HyperNeutrosophic Sets [37], Plithogenic Sets [38,39,40], Graphs [41], HyperGraphs [42,43], and SuperHyperGraphs [44,45].