CRISP SETS EXTENDED TO SOFT SETS AND HYPERSOFT SETS

ABSTRACT
This article presents a structured account of crisp, soft and hypersoft sets and develops a
reproducible hypothetical loan-assessment example. Parameterization and graded membership are
treated as distinct modelling choices. Crisp, fuzzy, intuitionistic fuzzy and single-valued
neutrosophic hypersoft mappings are defined on a common Cartesian product of attribute-value
sets. Four hypothetical applicants are described by income, credit history and collateral status.
Explicit categorical encodings and a weighted similarity function generate all fuzzy memberships; a
separate, disclosed information-completeness rule determines intuitionistic hesitation. The
neutrosophic illustration embeds those intuitionistic evaluations using the coordinate order truth,
indeterminacy and falsity. A pointwise fuzzy union is distinguished from the disjunctive
aggregation of different parameter fibres. Assessment against a specified desirable profile gives
scores of 1.00, 0.80, 0.40 and 0.00 for Applicants 1, 4, 2 and 3, respectively. Elementary properties
establish admissibility, recovery of exact profile matching and preservation of the ordering under
positive weights for these particular inputs. A one-parameter sensitivity calculation leaves the
illustrative action categories unchanged over the stated range. The contribution is a transparent
worked illustration of established frameworks, not a new hypersoft theory or an empirically
validated lending system. No claims of improved predictive accuracy or fairness are made.

Keywords: crisp set; soft set; hypersoft set; intuitionistic fuzzy set; single-valued neutrosophic set;
multi-criteria decision-making; hypothetical loan assessment.
AMS Mathematics Subject Classification 2020: Primary 03E72; Secondary 90B50.