On the Formal Foundations of D-Off Numbers and Neutrosophic
D-Numbers
Takaaki Fujita1,∗, Arif Mehmood2, Arkan A. Ghaib3
1Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan
2Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050,
KPK, Pakistan
3Department of Information Technology, Management Technical College, Southern Technical University,
Basrah, 61004, Iraq
Emails: Takaaki.fujita060@gmail.com; mehdaniyal@gmail.com; arkan.ghaib@stu.edu.iq
Abstract
A variety of uncertainty-handling frameworks—such as Fuzzy Sets,1 Hyperfuzzy Sets,2 Bipolar Fuzzy Sets,3
Neutrosophic Sets,4 Vague Set,5 Hesitant Fuzzy Sets,6, 7 Picture Fuzzy Sets,8 Soft Sets,9, 10 Rough Sets,11 and
Plithogenic Sets12, 13—have been extensively studied for modeling and reasoning under vagueness and impre-
cision. A fuzzy set extends classical set theory by assigning each element a membership value in the unit
interval [0, 1], thereby capturing partial inclusion.1 Neutrosophic Sets further generalize this idea by intro-
ducing three independent membership functions—truth, indeterminacy, and falsity—each mapping into [0, 1].
Many of these frameworks have been enriched by incorporating offset concepts, which permit membership
degrees to take values beyond the unit interval. Similarly, D-numbers extend Dempster–Shafer belief func-
tions by assigning to each subset B ⊆ X a mass D(B) ∈ [0, 1] with P
B D(B) ≤ 1, thus accommodating
incomplete uncertainty.14 In this work, we introduce and formally define four new constructs: D-OffNumber,
D-OverNumber, D-UnderNumber, and Neutrosophic D-Number, and we investigate their mathematical foun-
dations, structural properties, and interrelationships. The present study focuses exclusively on theoretical
development, leaving potential applications—such as their integration into decision-making frameworks—for
future research.
Keywords: Fuzzy Offset; Neutrosophic OffSet; Fuzzy Set; Neutrosophic Set; D-Number; Neutrosophic D-
Number