Ahmed, Muna Abduljabbar (2026) Fuzzy Functions: A Mathematical Framework and Applications Using MATLAB. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 7 (3). ISSN 2660-5309

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Abstract

Introduced by Zadeh in 1965, fuzzy set theory allows for the logical modeling of imprecision and linguistic uncertainty by generalizing classical bivalent logic to incorporate partial membership. The analytical features of fuzzy functions and fuzzy inference systems are examined with the rigor anticipated of classical real analysis in this study, which proposes a self-contained mathematical framework that treats them as two manifestations of a common algebraic structure. The framework begins with the algebraic foundations of fuzzy sets and their induced lattice order, proceeds to define fuzzy-valued functions (FVFs) via α-cuts and Zadeh’s extension principle, and establishes continuity, Hukuhara differentiability, and Kaleva integration within that setting. A worked MATLAB demonstration shows how the extension principle propagates a triangular fuzzy input through a nonlinear map, yielding a non-triangular output whose membership function is reconstructed from the α-cut family. Arithmetic on fuzzy numbers is treated through triangular, trapezoidal, and LR representations. Fuzzy inference systems (Mamdani and Takagi–Sugeno–Kang) are then derived as computable instances of FVFs: each system’s output surface is the centroid shadow of an underlying fuzzy-valued mapping, and its smoothness follows directly from the FVF continuity criterion. Universal approximation is established for both architectures. All theoretical constructs are accompanied by complete, reproducible MATLAB implementations, with three case studies— inverted-pendulum control, Mackey–Glass chaotic-series prediction, and Wisconsin breast-cancer classification—each explicitly connected back to the FVF formalism. Benchmark comparisons against classical and neural-network baselines demonstrate that, for low-dimensional problems with expert-available linguistic priors, well-tuned fuzzy models achieve competitive accuracy at an order-of-magnitude lower training cost.

Item Type:	Article
Uncontrolled Keywords:	fuzzy sets, membership functions, extension principle, fuzzy arithmetic, Mamdani FIS, Takagi–Sugeno–Kang FIS, matlab, fuzzy control, approximation theory, α-cuts
Subjects:	H Social Sciences
Depositing User:	admin eprints
Date Deposited:	30 Jul 2026 15:49
Last Modified:	30 Jul 2026 15:49
URI:	http://eprints.umsida.ac.id/id/eprint/16858

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