THE RING AND ALGEBRA OF INTUITIONISTIC SETS

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1 Hacettepe Journal of Mathematcs and Statstcs Volume , THE RING AND ALGEBRA OF INTUITIONISTIC SETS Alattn Ural Receved 01:08 :2009 : Accepted 19 :03 :2010 Abstract The am of ths study s to defne ntutonstc rngs, ntutonstc sgma rngs, ntutonstc algebras and ntutonstc sgma algebras, and on the other hand to adapt some well known rng and algebra theorems n classcal sets to ntutonstc sets. Keywords: Intutonstc algebra, Intutonstc rng AMS Classfcaton: 08 A 99, 03 E 99, 16 W Introducton After the ntroducton of the concept of fuzzy set by Zadeh [3], the dea of ntutonstc fuzzy sets was gven by Krassmr T. Atanassov [1], and some notes on ntutonstc sets and ntutonstc ponts were gven by Doğan Çoker [2]. In ths paper, the concepts of ntutonstc rng IR, ntutonstc sgma σ rng ISR, ntutonstc algebra IA and ntutonstc σ-algebra ISA are frst defned. On the other hand the valdty of some sutable theorems of rng and algebra theory n classcal sets for ntutonstc sets has been nvestgated and some theorems are frst gven. 2. Prelmnares 2.1. Defnton. [2] Let X be a nonempty fxed set. An ntutonstc set A s an object havng the form A = x, A 1, A 2, where A 1 and A 2 are subsets of X satsfyng A 1 A 2 =. The set A 1 s called the set of members of A, whle A 2 s called the set of nonmembers of A Defnton. [2] Let X be a nonempty fxed set and A, B ntutonstc sets of the form A = x, A 1, A 2 and B = x, B 1, B 2. Then a A B = x, A 1 B 1, A 2 B 2, b A B = x, A 1 B 1, A 2 B 2, Mehmet Akf Ersoy Unversty, Faculty of Educaton, Elementary Educaton Department, Elementary Mathematcs Educaton Programme, Burdur, Turkey. E-mal: altnurl@gmal.com

2 22 A. Ural c A B = A B, d A = x, A 2, A 1, e A B A 1 B 1, A 2 B Defnton. [2] = x,, X and X = x, X, Theorem. Let {A : J} be an arbtrary famly of ntutonstc sets n X, where A = x, A 1, A 2, and let B = x, B 1, B 2 be a fxed ntutonstc set n X. Then: B A = B A, B A = B A, B A = B A, v B A = B A. Proof. The unon of the sets A s A = x, A1, A2. Hence, B A = x, B 1, B 2 A 1 A 2 = x, B 1 A 1, B 2 A 2 = x, B1 A 1, B 2 A 2 = B A Smlar to. B A = B A = x, B 1, B 2 x, = = = = = x, B 1 x, B1 A 2 A 2, A 2, B 2, B2 A 1 x, B1 A 2, B 2 A 1 A 1 2 x, B1, B 2 x, A, A 1 B A A 1 v Smlar to. 3. Intutonstc rngs and ntutonstc algebras 3.1. Defnton. Let X be a nonempty fxed set and Ω any famly of ntutonstc sets n X. If the famly Ω satsfes the followng condtons, then the famly Ω s called a rng of ntutonstc sets n X, or for short an ntutonstc rng IRX. For all A Ω, B Ω we have A B Ω. For all A Ω, B Ω we have A B Ω.

3 The Rng and Algebra of Intutonstc Sets 23 If An IRX for all An IRX then IRX s called a σ-rng of ntutonstc sets or shortly an ntutonstc σ-rng ISRX. n By the defnton, for all A IRX and for every n N +, A IRX s =1 obvous Defnton. Let X be a nonempty fxed set and Θ any famly of ntutonstc sets n X. If the famly Θ satsfes the followng condtons then the famly Θ s called an algebra of ntutonstc sets n X, or shortly an ntutonstc algebra IAX. If A, B Θ then A B Θ, A Θ for each A Θ. If A n IAX for all A n IAX, then IA s called a σ-algebra of ntutonstc sets, or shortly an ntutonstc σ-algebra ISAX Theorem. Every IAX s also an IRX. Proof. Let A,B IAX. Thus by Defnton 3.2, we have A IAX, A B IAX and A B IAX. For A B = A B = A B we have A B IAX. So, by the defnton of IRX, IAX s also an IRX Theorem. Every ISAX s also an ISRX. Proof. Obvous by Theorem 3.3 and Defntons 3.1, Theorem. If X IRX then IRX s an IAX, and also f X ISRX then ISRX s an ISAX. Proof. Let X, A IRX. Then, by the equalty A = X A we have A IRX. Ths completes the proof Theorem. Let A =1 be a countable famly of ntutonstc sets n an ISAX. Then A ISAX. =1 Proof. By the defntons; For each A ISAX, A ISAX and By the equalty A = A we have =1 =1 A ISAX and =1 A ISAX. =1 A ISAX. = Theorem. Let X be a nonempty fxed set and Φ a famly of ntutonstc sets n X. In ths case, there s a mnmal ISAX contanng the famly Φ. Ths famly s called the Φ-produced mnmal ISAX. Proof. Let = {ISAX : ISAX contans the famly Φ} and defne ISAX = IAX. IAX Now let us prove that the famly ISAX s an ISAX.

4 24 A. Ural A,B ISAX = A, B ISAX ISAX = A B ISAX ISAX = A B ISAX. A ISAX = A ISAX ISAX = A ISAX ISAX = A ISAX. We must prove that A n ISAX for each A n ISAX, n N +. For each n N + and A n ISAX, A n ISAX for each ISAX. For each n N + and ISAX we have A n ISAX. Hence, A n ISAX Theorem. Let IA 1X and IA 2X be ntutonstc algebras n X. Then, IA 1X IA 2X s an ntutonstc σ-algebra n X. Proof. For every A,B IA 1X IA 2X, we must prove that A B IA 1X IA 2X. A, B IA 1X IA 2X = A,B IA 1X and A, B IA 2X = A B IA 1X and A B IA 2X = A B IA 1X IA 2X. For every A IA 1X IA 2X, we must prove that A IA 1X IA 2X. A IA 1X IA 2X = A IA 1X and A IA 2X = A IA 1X and A IA 2X = A IA 1X IA 2X. For A n IA 1X IA 2X, n N +, we have A n IA 1X IA 2X snce IA 1X and IA 2X are ntutonstc σ-algebras. From, and, IA 1X IA 2X s an ntutonstc σ-algebra. { } 3.9. Corollary. Ω = s an ntutonstc rng. Proof. For = x,,x, = x,, X X = x,, X = Ω. = = x,, X x, X, = x, X, X = x,,x = Ω. { } By and, Ω = s an ntutonstc rng Corollary. For X, let P X be the famly of all ntutonstc sets n X. Then, P X s an ISAX.

5 The Rng and Algebra of Intutonstc Sets 25 Proof. Let A = x, A 1, A 2 and B = x,b1, B 2 be any two ntutonstc sets n X. A B = x, A 1 B 1, A 2 B 2 P X, A = x, A 2, A 1 P X, For A n = x,a 1 n, A 2 n P X, n N +, A n = x,a 1 n, A 2 n = x, A 1 n, A 2 n P X Example. Let A be an ntutonstc set n X of the form A = x,a 1,. Then M = { A, A } s an ISAX. Proof. A A = x,a 1, x,, A 1 = x, A 1, A 1 = x,a 1, = A M, A M and A = A M Example. { Let } X and A = x, A 1, be an ntutonstc set n X. In ths case, N = A,A,, X s an ISAX. Proof. The unon of any two ntutonstc sets n N s also an ntutonstc set n N: A A = A, A = A, A X = X, A = A, A X = X. The complement of a ntutonstc set n N s also an ntutonstc set n N: = X, X =, A = A. X A A = X. From,, we have the famly N s an ISAX Example. Let A be an ntutonstc set havng the form A = x, A 1, A 2. The famly P = { A, A, A A, A A } s an ISAX. Proof. The unon of any two ntutonstc sets n P s also an ntutonstc set n P: A A A A = A A, A A A = A, A A A = A. The complement of an ntutonstc set n P s also an ntutonstc set n P: A A = A A, A A = A A. A countable unon of ntutonstc sets n P s also an ntutonstc set n P: A A A A A A = A A. From,, we have that the famly P s an ISAX. 4. Conclusons In ths paper we have adapted some rng and algebra theorems n classcal sets to ntutonstc sets. Other theorems couldn t been adapted because not all of the well known propertes of the operatons unon, ntersecton, dfference, complement for classcal sets are not vald for ntutonstc sets. One of the man reasons s that the unon of an ntutonstc set and ts complement does not always equal the ntutonstc unversal set X, whch contans all of other ntutonstc sets. Another reason s that the ntersecton of an ntutonstc set and ts complement always does not always equal the empty set, whch s subset all of other empty sets.

6 26 A. Ural References [1] Atanassov, K., Intutonstc Fuzzy Sets VII ITKR s Sesson, Sofa, [2] Çoker, D. A Note on ntutonstc sets and ntutonstc ponts, Turksh Journal of Mathematcs 203, , [3] Zadeh, L.A., Fuzzy sets, Informaton and Control. 8, , 1965.