ChatClassical fuzzy sets struggled to model real-world problems containing active contradictions and indeterminate information; single-valued neutrosophic relations provide a rigorous mathematical framework to compute truth, indeterminacy, and falsity independently.
In medical diagnosis, expert systems, and automated decision-making, real-world information is rarely crisp binary truth: observations are frequently incomplete, contradictory, or shrouded in neutral doubt.
Classical logic demands that statements are either true or false, while Zadeh's fuzzy logic models degrees of truth but forces truth and falsity into rigid complementary bounds that cannot represent genuine logical indeterminacy.
This mathematical treatise investigates single-valued neutrosophic relations (SVNRs), which independently quantify degrees of truth-membership, indeterminacy-membership, and falsity-membership. The author proves fundamental composition theorems, symmetry properties, and transitive closures for relational structures.
These neutrosophic relational theorems establish the rigorous mathematical foundation required for next-generation multi-criteria decision support systems, medical diagnostic AI, and automated risk analysis in uncertain environments.
On single valued neutrosophic relations
Smarandache initiated neutrosophic sets (NSs) which can be used as a mathematical tool for dealing with indeterminate and inconsistent information. In order to apply NSs conveniently, single valued neutrosophic sets (SVNSs) were proposed by Wang et al. In this paper, we propose single valued neutrosophic relations (SVNRs) and study their properties. The notions of anti-reflexive kernel, symmetric kernel, reflexive closure, and symmetric closure of a SVNR are introduced, respectively. Their accurate calculate formulas and some properties are explored. Some examples are also given. Finally, single valued neutrosophic relation mappings and inverse single valued neutrosophic relation mappings are introduced, and some interesting properties are also obtained.
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