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arxiv: 2605.14560 · v3 · pith:Z54CRAAInew · submitted 2026-05-14 · 🧮 math.GM

Roughness and entropy measures of a soft set

Santanu Acharjee , Sankar K. Pal This is my paper

Pith reviewed 2026-05-21 08:57 UTC · model grok-4.3

classification 🧮 math.GM
keywords soft setsroughness measuresentropy measuresuncertaintysoft computingrough setsMolodtsov
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The pith

Soft sets can be assigned two new roughness measures and six entropy measures while preserving their original definition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines two distinct roughness measures for soft sets within separate conceptual frameworks. It also introduces six entropy measures to quantify uncertainty in these structures. Properties of the measures receive both theoretical proofs and computational checks. All developments stay inside Molodtsov's original soft set axioms, and the work is contrasted with classical rough set theory to show its distinct contributions to roughness characterization.

Core claim

Two roughness measures and six entropy measures are introduced for soft sets. These are investigated for their properties through theoretical analysis and computational techniques. The framework is shown to be novel with respect to roughness characterization while strictly preserving the foundational principles of soft set theory established by Molodtsov, and a comparison with classical rough set theory is provided.

What carries the argument

The two roughness measures defined in distinct conceptual frameworks together with the six entropy measures, which quantify uncertainty while respecting the attribute-parameterized structure of soft sets.

If this is right

  • The measures support systematic theoretical and computational study of roughness in soft sets.
  • Entropy functions provide quantifiable uncertainty assessments that remain consistent with soft set axioms.
  • Comparative distinctions from classical rough sets clarify the added value of the attribute-oriented approach.
  • The framework can be applied directly in domains already using soft sets for decision problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These measures could be tested on real attribute datasets from social sciences to check whether they produce more stable rankings than existing soft set methods.
  • Integration with hybrid models that combine soft sets with other uncertainty calculi becomes possible once the new entropy values are available.
  • The computational validation steps in the paper suggest a route for implementing the measures in software for larger parameter spaces.

Load-bearing premise

The foundational principles of soft set theory as established by Molodtsov are strictly preserved throughout the development of the measures.

What would settle it

A concrete soft set example in which one of the proposed roughness measures violates monotonicity under inclusion of the parameter sets or fails to reduce to a known rough set case when the soft set is crisp would disprove the claims.

Figures

Figures reproduced from arXiv: 2605.14560 by Sankar K. Pal, Santanu Acharjee.

Figure 1
Figure 1. Figure 1: Upper soft approximation of a soft set ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Upper soft approximation of a soft set ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Lower soft approximation of a soft set ( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Lower soft approximation of a soft set ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Boundary of a soft set (S, A) determined over an approximation space (X, R). In [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Boundary of a soft set (S, A) determined over an approximation space (X,R). In [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Graphs of accuracy measures of ρ P Ri (S, A) and ρ Y Ri (S, A) for all equivalence relations Ri , where i ∈ {1, 2, 3, ..., 15}, on X = {a, b, c, d} ; (b) Graphs of accuracy measures of ρ P Ri (C(S, A)) and ρ Y Ri (C(S, A)) for all equivalence relations Ri , where i ∈ {1, 2, 3, ..., 15} on X = {a, b, c, d}. 5 Entropy of a soft set From the above section, it is observed that the accuracy measure of a sof… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Graphs of accuracy measures of ρ P Ri (S, A) and ρ Y Ri (S, A) for all equivalence relations Ri , where i ∈ {1, 2, 3, ..., 15}, on X = {a, b, c, d} ; (b) Graphs of accuracy measures of ρ P Ri (C(S, A)) and ρ Y Ri (C(S, A)) for all equivalence relations Ri , where i ∈ {1, 2, 3, ..., 15} on X = {a, b, c, d}. more recently, Tang et al. [41] addressed this limitation and proposed an uncertain entropy measu… view at source ↗
Figure 5
Figure 5. Figure 5: 3D surface plot of Ent1P e (S, A) with its maximum value obtained at the point (1/e, 1/e, 1). Theorem 5.2. Let (X, R1) and (X, R2) be two approximation spaces, and (S, A) be a soft set defined over X. If R1 ⪯ R2, then (Ent1P e (S, A))R2 ≤ (Ent1P e (S, A))R1 . Proof. If R1 ⪯ R2, by Corollary 4.1, θ P R1 (S, A) ≤ θ P R2 (S, A) and θ P R1 (C(S, A)) ≤ θ P R2 (C(S, A)). So, we get that θ P R1 (S, A) loge (θ P R… view at source ↗
Figure 5
Figure 5. Figure 5: 3D surface plot of Ent1P e (S, A) with its maximum value obtained at the point (1/e, 1/e, 1). Theorem 5.2. Let (X,R1) and (X,R2) be two approximation spaces, and (S, A) be a soft set defined over X. If R1 ⪯ R2, then (Ent1P e (S, A))R2 ≤ (Ent1P e (S, A))R1 . Proof. If R1 ⪯ R2, by Corollary 4.1, θ P R1 (S, A) ≤ θ P R2 (S, A) and θ P R1 (C(S, A)) ≤ θ P R2 (C(S, A)). So, we get that θ P R1 (S, A) loge (θ P R1 … view at source ↗
Figure 6
Figure 6. Figure 6: 3D surface plot of Ent2P e (S, A) with its maximum value obtained at the point (1, 1, 1) when β = e. Theorem 5.4. Let (X, R1) and (X, R2) be two approximation spaces, and (S, A) be a soft set defined over X. If R1 ⪯ R2 and β is fixed, then (Ent2P β (S, A))R2 ≤ (Ent2P β (S, A))R1 . Proof. The proof can be obtained by following the proof of Theorem 5.2 and keeping β fixed. Theorem 5.5. Let (X, R) be an appro… view at source ↗
Figure 6
Figure 6. Figure 6: 3D surface plot of Ent2P e (S, A) with its maximum value obtained at the point (1, 1, 1) when β = e. Theorem 5.4. Let (X,R1) and (X,R2) be two approximation spaces, and (S, A) be a soft set defined over X. If R1 ⪯ R2 and β is fixed, then (Ent2P β (S, A))R2 ≤ (Ent2P β (S, A))R1 . Proof. The proof can be obtained by following the proof of Theorem 5.2 and keeping β fixed. Theorem 5.5. Let (X,R) be an approxim… view at source ↗
Figure 7
Figure 7. Figure 7: 3D surface plot of EntP e (S, A) with its maximum value obtained at the point (1, 1, 1) when β = e. Theorem 5.7. Let (X, R) be an approximation space, and (S, A) be a soft set defined over X. If β1 < β2, then EntP β1 (S, A) ≤ EntP β2 (S, A). Proof. The proof can be obtained easily. Thus, we skip it. Since we discussed that the accuracy measure of a soft set in the sense of Pawlak is not following the axiom… view at source ↗
Figure 7
Figure 7. Figure 7: 3D surface plot of EntP e (S, A) with its maximum value obtained at the point (1, 1, 1) when β = e. Theorem 5.7. Let (X,R) be an approximation space, and (S, A) be a soft set defined over X. If β1 < β2, then EntP β1 (S, A) ≤ EntP β2 (S, A). Proof. The proof can be obtained easily. Thus, we skip it. Since we discussed that the accuracy measure of a soft set in the sense of Pawlak is not following the axioms… view at source ↗
Figure 8
Figure 8. Figure 8: 2D plot of [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: 2D plot of Ent3P e (S, A) with its maximum value obtained at the point (1/e, 1/e). Theorem 5.9. Let (X,R1) and (X,R2) be two approximation spaces, and (S, A) be a soft set defined over X. If R1 ⪯ R2, then (Ent3P e (S, A))R2 ≤ (Ent3P e (S, A))R1 . Proof. Since R1 ⪯ R2, we get θ Y R1 (S, A) ≤ θ Y R2 (S, A). Hence, loge (θ Y R1 (S, A)) ≤ loge (θ Y R2 (S, A)). Thus, we have (Ent3P e (S, A))R2 ≤ Ent3P e (S, A))… view at source ↗
Figure 9
Figure 9. Figure 9: 2D plot of Ent4P β (S, A) with its maximum value obtained at the point (1, 1). Theorem 5.12 proves the basic properties of Ent′P β (S, A). Also, Ent′P β (S, A) is strictly increasing if β is strictly increasing [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: 2D plot of Ent4P β (S, A) with its maximum value obtained at the point (1, 1). Theorem 5.12 proves the basic properties of Ent′P β (S, A). Also, Ent′P β (S, A) is strictly increasing if β is strictly increasing [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 2D plot of Ent′P β (S, A) with its maximum value obtained at the point (1, 1). 6 Comparisons of six entropy measures of a soft set In Section 5, we introduced six types of entropy measures for a soft set. It is therefore natural to examine the significance of these different notions and to explore which one is more suitable. In this section, we present the following table to highlight the distinctions amo… view at source ↗
Figure 10
Figure 10. Figure 10: 2D plot of Ent′P β (S, A) with its maximum value obtained at the point (1, 1). 29 [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Region bounded by red color, blue color and green color in an approximation space [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 11
Figure 11. Figure 11: Region bounded by red color, blue color and green color in an approximation space ( [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Lower approximation using Pawlak’s rough set. [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: In this case as well, we do not have the flexibility to consider the overlap between the two regions [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 13
Figure 13. Figure 13: Upper approximation using Pawlak’s rough set. [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 12
Figure 12. Figure 12: Lower approximation using Pawlak’s rough set. [PITH_FULL_IMAGE:figures/full_fig_p033_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Lower and upper approximations using of rough set theory. [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: Upper approximation using Pawlak’s rough set. [PITH_FULL_IMAGE:figures/full_fig_p033_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Lower and upper approximations using of rough set theory. [PITH_FULL_IMAGE:figures/full_fig_p034_14.png] view at source ↗
read the original abstract

Soft set theory is an important and emerging area within soft computing, owing to its attribute-oriented mathematical framework and its wide applicability in diverse domains, including science and social sciences. The theoretical constraints associated with the selection of subsets of the sets of attributes in soft set theory have further motivated the development of hybrid and extended theoretical models. In this paper, we introduce two distinct roughness measures and six entropy measures for soft sets and systematically investigate their properties using both theoretical analysis and computational techniques. The proposed roughness measures are defined within two distinct conceptual frameworks. Throughout the development of these measures and the corresponding results, the foundational principles of soft set theory, as established by Molodtsov, are strictly preserved. Furthermore, the proposed framework is shown to be novel with respect to roughness characterization, and a comparative analysis with classical rough set theory is presented to highlight the theoretical distinctions and contributions of this work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces two distinct roughness measures and six entropy measures for soft sets. It systematically investigates their properties using theoretical analysis and computational techniques while strictly preserving the foundational principles of soft set theory as established by Molodtsov. A comparative analysis with classical rough set theory is presented to highlight theoretical distinctions and the novelty of the roughness characterization.

Significance. If the definitions, properties, and computational verifications hold as described, the work could provide useful extensions for uncertainty quantification in soft computing applications such as decision-making under attribute-based uncertainty. The explicit preservation of Molodtsov's axioms and the side-by-side comparison with rough sets are strengths that help position the contribution as compatible with existing soft-set literature while offering distinct roughness tools.

minor comments (2)
  1. Abstract: the repeated reference to 'theoretical analysis and computational techniques' could be streamlined to improve conciseness without loss of meaning.
  2. The manuscript would benefit from a brief explicit statement (perhaps in the introduction or a dedicated subsection) of the specific computational methods or software employed to verify the proposed measures, to aid reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. We appreciate the recognition that our work preserves Molodtsov's foundational principles while offering novel roughness and entropy measures for soft sets, along with the comparison to classical rough set theory.

Circularity Check

0 steps flagged

No significant circularity; measures defined independently within Molodtsov framework

full rationale

The paper introduces two roughness measures and six entropy measures for soft sets while explicitly preserving Molodtsov's foundational axioms (external reference, not self-citation). Properties are investigated via theoretical analysis and computational techniques, with explicit comparison to classical rough set theory to highlight distinctions. No equations or definitions reduce by construction to fitted inputs, prior self-citations, or renamed known results. The derivation chain remains self-contained against external benchmarks, with novelty claimed through direct contrast rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or additional axioms beyond the stated preservation of Molodtsov's principles are identifiable.

axioms (1)
  • domain assumption Foundational principles of soft set theory as established by Molodtsov are strictly preserved.
    Stated directly in the abstract as a requirement maintained throughout the development of the measures.

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