Abstract
In this paper, a new hypothesis of fuzzy number has been proposed which is more precise and direct. This new proposed approach is considered as an equivalence class on set of real numbers R with its algebraic structure and its properties along with theoretical study and computational results. Newly defined hypothesis provides a well-structured summary that offers both a deeper knowledge about the theory of fuzzy numbers and an extensive view on its algebra. We defined field of newly defined fuzzy numbers which opens new era in future for fuzzy mathematics. It is shown that, by using newly defined fuzzy number and its membership function, we are able to solve fuzzy equations in an uncertain environment. We have illustrated solution of fuzzy linear and quadratic equations using the defined new fuzzy number. This can be extended to higher order polynomial equations in future. The linear fuzzy equations have numerous applications in science and engineering. We may develop some iterative methods for system of fuzzy linear equations in a very simple and ordinary way by using this new methodology. This is an innovative and purposefulness study of fuzzy numbers along with replacement of this newly defined fuzzy number with ordinary fuzzy number.
Highlights
In the current scenario, many real life problems cannot be dealt with the classical set theory and a need arises of its extension
In 1960’s an eminent scientist Zadeh [1] proposed a novel theory known as Fuzzy Sets, a generalization of classical set theory
The use of fuzzy numbers is used in solving fuzzy equations [4, 7,8,9,10], fuzzy calculus [11,12], fuzzy graph [13], fuzzy differential equations [14,15], and many other areas of pure and applied sciences
Summary
We define fuzzy algebraic operations +, −, × and / on the set. For Rε = {rr = [r r ]ε , rr ∈ R }, let r r1, r r2 ∈ Rε, we define the algebraic operations as follows: 1) r r1 + r r2 = [ r r1 ]ε + [r r2 ]ε = [ rr1 + rr2 ]ε = rr 1 + r r2 where r r1 + r r2 = [rr, ε1, δδ, λλ1, γγ1] + [rr2 + ε2, δδ, λλ2, γγ2] =. For rr, rr2 ∈ R, we define the algebraic operations of addition and multiplication on Rε as follows:. R r1 + r r2 = [ r r1 ]ε + [r r2 ]ε = [ rr1 + rr2 ]ε = rr 1 + r r2 r r1 × r r2 = [ r r1 ]ε × [ r r2 ]ε = [ r r1rr2 ]ε = r r 1 r r2. Form the above results, we conclude that there exists an algebraic structure on Rε
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