Abstract
Overtime, mathematics had been used as a tool in modeling real life phenomenon. In some cases, these problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty. The uncertainty could either be due to lack of clear boundaries in the description of the object or perhaps due to randomness. In this article, we study a mathematical tool discovered in 1965 by Zadeh suitable for modeling real life phenomenon and examined operations on such a tool. Motivated by the work of Zadeh, we studied operators on Type-1 Fuzzy Sets (T1FSs) and Type-2 Fuzzy sets (T2FSs) and provided examples, one of which is a variant of the Yager complement function for which the complement operator was graphically illustrated. The joint and the meet operators were also studied and examples provided. Non-standard operators were defined on T1FSs and T2FSs and also classified into two groups; the triangular-norm (t-norm) and triangular-conorm (t-conorm). Using t-norm and t-conorm, an example was adopted from Castillo and Aguilar to illustrate the computation of the standard operation on T2FSs. Finally, future research direction was provided based on what is yet to be achieved in fuzzy set theory.
Highlights
Overtime, mathematics had been used as a tool in modeling real life phenomenon
These problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty
In the study of Zadeh [18] the standard operations on fuzzy sets were first defined. These were the union operations, intersection and complement operations on fuzzy sets, a unique concept in fuzzy set theory without its replica in the Cantorian theory. It has been established in the study of Chen and Zadeh [4] that operations on fuzzy sets has no counterpart in the classical set theory in particular and in classical mathematics in general
Summary
Mathematics had been used as a tool in modeling real life phenomenon. In some cases, these problems cannot fit-into the classical deterministic or stochastic modeling techniques, perhaps due system complexity arising from lack of complete knowledge about the phenomenon or some uncertainty. In the study of Zadeh [18] the standard operations on fuzzy sets were first defined These were the union operations, intersection and complement operations on fuzzy sets, a unique concept in fuzzy set theory without its replica in the Cantorian theory. Mathematician have extended operations on T1FSs to T2FSs as noticed in [10,11,12, 14, 15] and recently [3, 8] and the references therein This fact necessitated the need for further research on the operations on type-1 and type-2 fuzzy sets, the subject of the present paper.
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