More on Cartesian Products over Intuitionistic Fuzzy Sets

Transcription

1 International Mathematical Forum, Vol. 7, 01, no. 3, More on Cartesian Products over Intuitionistic Fuzzy Sets Annie Varghese 1 Department of Mathematics St. Peter s College, Kolencherry Kerala, India anniestpc@gmail.com Sunny Kuriakose Principal, BPC College, Piravom, Kerala, India Abstract. In this paper we introduce two versions of Cartesian products over Intuitionistic Fuzzy Sets and study their properties. Keywords: Intuitionistic Fuzzy Sets, Cartesian product over Intuitionistic Fuzzy Sets 1 Introduction Atanassov [1] has defined five versions of cartesian products of two Intuitionistic Fuzzy Sets (IFSs) 1,, 3, 4, 5. Also, Velin Andonov [] introduced a sixth version 6 of cartesian product of two IFSs. In an earlier paper by the authors, three different types of Cartesian products 7, 8 and 9 are introduced. In this paper we introduce two more versions of cartesian products 10 and 11 of two IFSs. Let E 1 and E be two universes and let A = { x, μ A (x),ν A (x) x E 1 } and B = { y, μ B (y),ν B (y) y E } be two IFSs: A over E 1 and B over E. Definition 1.1. The five cartesian products of two IFSs A and B are defined 1 Correspondence address: Manithottathil, S. Marady P.O., Muvattupuzha, Kerala , India.

2 1130 A. Varghese and S. Kuriakose as follows: A 7 B = { x, y, min(1,μ A (x)+μ B (y)), max(0,ν A (x)+ν B (y) 1) x E 1,y E } A 8 B = { x, y, max(0,μ A (x)+μ B (y) 1), min(1,ν A (x)+ν B (y)) x E 1,y E } A 9 B = { x, y, μ A (x)μ B (y), ν A (x)ν B (y) x E 1,y E } μ A (x)μ B (y) A 10 B = { x, y, μ A (x)+μ B (y), ν A (x)ν B (y) ν A (x)+ν B (y) x E 1,y E } for which we will accept that if μ A (x) =μ B (y) = 0, then μ A(x)μ B (y) = 0 and μ A (x)+μ B (y) if ν A (x) =ν B (y) = 0, then ν A(x)ν B (y) =0. ν A (x)+ν B (y) μ A (x)+μ B (y) A 11 B = { x, y, (μ A (x)μ B (y)+1), ν A (x)+ν B (y) (ν A (x)ν B (y)+1) x E 1,y E } Result 1.1. [4] A 7 B, A 8 B and A 9 B are IFS over E 1 E where is the classical cartesian product on ordinary sets E 1 and E. Result 1.. A 10 B is an IFS over E 1 E. Proof. If μ A (x)+μ B (y) > 0 and ν A (x)+ν B (y) > 0 then 0 μ A(x)μ B (y) μ A (x)+μ B (y) + ν A(x)ν B (y) ν A (x)+ν B (y) μ A(x)+μ B (y) + ν A(x)+ν B (y) 1 Result 1.3. A 11 B is an IFS over E 1 E. Proof. For four real numbers 0 a, b, c, d 1, a + c ac +1. a + c Then (ac +1) 1 (If a+c >ac+1, then a+c(1 a) > 1, a contradiction). Similarly 0 a + c (ac +1) + b + d (bd +1) 1. From these inequations it follows that A 11 B is an IFS Definition 1.. [1] If A is the IFS { x, μ A (x),ν A (x) x E}, then Ā = { x, ν A (x),μ A (x) x E}. b+d 1. (bd+1)

3 Cartesian products over intuitionistic fuzzy sets 1131 Properties Proposition.1. Let E 1 and E be two universes. If A and B are IFSs over E 1 and C is an IFS over E, then the following equalities hold where { 9, 10 } (A B) C=(A C) (B C) (1) (A B) C=(A C) (B C) () C (A B)=(C A) (C B) (3) C (A B)=(C A) (C B) (4) Proof. We shall prove (1) for 9 and (1) for 10. A B= { x. max(μ A (x),μ B (x)), min(ν A (x),ν B (x)) x E 1 } C = { y, μ C (y),ν C (y) y E } Proof (1) for 9. Let be the cartesian product 9 (A B) C = { x, y, max(μ A (x),μ B (x))μ C (y), min(ν A (x),ν B (x))ν C (y) x E 1,y E } (A C) (B C)={ x, y, max( μ A (x),μ C (y), μ B (x),μ C (y)), min( ν A (x),ν C (y), ν B (x),ν C (y)) x E 1,y E }. Assume that μ A (x) >μ B (x) μ A (x)μ C (y) >μ B (x)μ C (y) μa (x)μ C (y) > μ B (x)μ C (y) μ (A B) C (x, y) = μ A (x)μ C (y) =μ (A C) (B C) (x, y). Similarly we can prove that ν (A B) C (x, y) =ν (A C) (B C). The proof of () for 9 is similar. Proof (1) for 10. Let be the cartesian product 10. Assume μ A (x) >μ B (x). μ (A B) C (x, y) = max(μ A(x),μ B (x))μ C (y) max(μ A (x),μ B (x)+μ C (y) = μ A(x)μ C (y)

4 113 A. Varghese and S. Kuriakose μ (A C) (x, y) = μ A(x)μ C (y) μ (B C) (x, y) = μ B(x)μ C (y) μ B (x)+μ C (y) ( ) μa (x)μ C (y) μ (A C) (B C) (x, y) = max, μ B (x)μ C (y) μ B (x)+μ C (y) = μ A(x)μ C (y) since if μ A (x) >μ B (x), μ A (x) μ C (y) > μ B (x) μ C (y) Similar is the case for non-membership grade also. The proof of () for 10 is similar. (3) and (4) are direct corollaries of (1) and () Remark.1. μ A C (x, y) =μ C A (y, x) and ν A C (x, y) =ν C A (y, x) where is any of 7, 8, 9, 10, 11. Proof. μ C 11 A(y, x) = μ C(y)+μ A (x) (μ C (y)μ A (x)+1) = μ A 11 C(x, y) Similar is the case for non membership grade also. Similar result holds for 7, 8, 9 and 10. Proposition.. [4] If A is an IFS over E 1 and B is an IFS over E, then Ā 7 B = A 8 B Ā 8 B = A 7 B Ā 9 B = A 9 B Proposition.3. Formulas similar to DeMorgan s law hold for the cartesian products 10 and 11.IfA is an IFS over E 1 and B is an IFS over E, then Ā 10 B = A 10 B Ā 11 B = A 11 B Therefore, the cartesian products 10 and 11 are autodual. Proof. Ā = { x, ν A (x),μ A (x) x E 1 } B = { y, ν B (y),μ B (y) y E }

5 Cartesian products over intuitionistic fuzzy sets 1133 μ A (x)μ B (y) Ā 10 B = { x, y, μ A (x)+μ B (y), ν A (x)ν B (y) ν A (x)+ν B (y) x E 1,y E } = A 10 B. Similarly the other result can be proved. 3 Conclusion In this paper we have introduced two versions of cartesian products of two IFSs and we have proved some of their properties. We have proved some properties of other versions of cartesian products also. References [1] K. Atanassov, Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, Chapter 1 (1999). [] Velin Andonov, On some properties of one Cartesian product over Intuitionistic fuzzy sets, Notes on Intuitionistic Fuzzy Sets, Vol. 14 (008), No.1, [3] K. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, Vol. 0(1986), No. 1, [4] Annie Varghese, Sunny Kuriakose, Cartesian products over Intuitionistic Fuzzy Sets, Accepted for publication in International Journal of Fuzzy Mathematics and Systems. Received: October, 011