Fuzzy Sequences in Metric Spaces

Transcription

1 Int. Journal of Math. Analysis, Vol. 8, 2014, no. 15, HIKARI Ltd, Fuzzy Sequences in Metric Spaces M. Muthukumari Research scholar, V.O.C. College, Tuticorin, India A. Nagarajan Department of Mathematics, V.O.C. College, Tuticorin, India M. Murugalingam Department of Mathematics, Thiruvalluvar College Papanasam , India Copyright 2014 M. Muthukumari, A. Nagarajan and M. Murugalingam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We introduce the new concept fuzzy sequences in a metric space. We define a new convergence property. 1. Introduction In a metric space sequence is a good tool to study the important properties. The closure of a set A can be characterised using convergent sequences in A. The continuity of a function from one metric space to another can be characterised using convergent sequences. In the year 1965 Lotfi A.Zadeh [2] introduced the concept of fuzzy sets. So far Mathematicians have been using the Binary logic. But people realised the inadequacy of binary logic in many situations. Hence the multiway logic namely fuzzy logic was introduced by Zadeh. Using the concept of fuzzy, Mathematicians developed Fuzzy algebra, Fuzzy topology and many more new concepts. In the year 1968 C.L.Chang [1] introduced Fuzzy topological spaces.

2 700 M. Muthukumari, A. Nagarajan and M. Murugalingam In this paper we introduce fuzzy sequence in a metric space and also we introduce a new convergent concept which is an extension of already existing crisp concept. Keywords : Fuzzy sequence, Fuzzy convergence 2. Fuzzy sequences Definition 1: Let M be a non empty set. A fuzzy set A on N Χ M is called a fuzzy sequence in M. i.e., A: N Χ M > [0,1] is called a fuzzy sequence in M. Example 1: 1. Take M = N. Define A: N X N > [0,1] as A(n,x) = 1/ (n+x) for all n є N, for all x ε N. Clearly A is a fuzzy sequence in N. Remark 1: After defining fuzzy sequence in M, we see the difference between a sequence in M and a fuzzy sequence in M. A sequence in M is a function from N to M, whereas a fuzzy sequence in M is a function from N X M >[0,1]. Hereafter ordinary sequence which is a function from N to M will be called as a crisp sequence. The link between these two is illustrated by the following theorem. Theorem 1: Every crisp sequence in M is a fuzzy sequence in M. Proof: Let M be a non empty set. Let S be the set of all crisp sequences on M and let F be the set of all fuzzy sequences on M. We consider a function t : S > F as follows. Take any f є S. Then f: N > X. Now define A f : N X M > [0,1] as A f (n,x) = 1 if f(n) = x. 0 otherwise.

3 Fuzzy sequences in metric spaces 701 Clearly A f is a fuzzy sequence in M. Let t(f) = A f. We claim that this function t is one one. Let f and g belong to S and let f g. Then there exists n 0 є N, such that f(n 0 ) g(n 0 ). Let f(n 0 ) = x and g(n 0 ) = y then x y. Now A f (n 0,x) = 1 and A g (n 0,x) = 0. Hence t(f) t(g). Therefore t is one one. Hence every crisp sequence in M is uniquely associated to a fuzzy sequence in M. Therefore every crisp sequence is a fuzzy sequence. Result 1: Converse of the theorem is not true. There are fuzzy sequences which are not crisp sequences. This is shown by the following example. Example 2: Let M = Z. Define A : N X M > [0,1] as A(n,x) = 1/(n+ l x l ). Clearly A is a fuzzy sequence but A cannot be considered as a crisp sequence. Result 2: We have seen that some fuzzy sequences are crisp sequences where some of them are not. We analyse the difference. We find conditions such that if a fuzzy sequence satisfies the conditions then it will be a crisp sequence. Theorem 2: Let M be a non empty set. A fuzzy sequence A on M is a crisp sequence if A satisfies the following conditions. 1. A(n,x) = 0 or 1 for all n є N, for all x є M. 2. For each n є N, there exists unique x in M such that A(n,x) = 1. Proof: Let A be a fuzzy sequence on M satisfying the given conditions. We claim that there exists a crisp sequence f on M such that A f = A. We define crisp sequence f on M where f(n) = x if A(n,x) = 1. Since for each n є N, there exists unique x in M such that A(n,x) = 1, f(n) is defined for all n є N. Hence f : N > M is a crisp sequence. Now we see that for this sequence f, A f = A. If A f (n,x) = 1 then f(n) = x. Hence A(n,x) = 1. Therefore A f (n,x) = 1 implies A f (n,x) = A(n,x). If A f (n,x) = 0 then f(n) x. Hence A(n,x) 1. Therefore A(n,x) = 0. Hence A f (n,x) = 0 implies A f (n,x) = A(n,x). Therefore there exists f: N > X such that A f = A. Hence A can be considered as a crisp sequence.

4 702 M. Muthukumari, A. Nagarajan and M. Murugalingam 3. Convergence We can define the convergence of a fuzzy sequence in a metric space. Here we introduce a concept namely level of convergence. Definition 2: Let (M,d) be a metric space and let A be a fuzzy sequence on M. Let α є (0,1]. Let a є M. A is said to converge to a at level α if 1. For each n є N, there exists atleast one x in M where A(n,x) α 2. Given ε >0, there exists n 0 є N such that d(x,a)< ε for all n n 0 and for all x with A(n,x) α ie., given ε > 0, there exists n 0 є N such that n n 0 and A(n,x) α implies d(x,a) < ε. We write A > a. Example 3: Consider R with usual metric. Define A : N X M >[0,1] as A(n,x) = 1 if x = 1/n and A(n,x)= 0 otherwise.we claim that A >0. Take any α > 0. α є (0,1] 1. For each n ε N, 1/n є M such that A(n,1/n) α 2. Let ε > 0 be given. Take n 0 є N such that n 0 > 1/ε Let n n 0 and A(n,x) α Now d(x,a) =l x a l = l x 0 l = l x l. n n 0 and A(n,x) α => n > 1/ε and A(n,x) = 1 => n > 1/ε and x = 1/n Now d(x,a) = l x l = l 1/n l = 1/n < ε. Given ε > 0, there exists n 0 є N, such that n n 0 and A(n,x) α => d(x,0) < ε. Hence A > 0.

5 Fuzzy sequences in metric spaces 703 Theorem 3: The concept of convergence of fuzzy sequence is an extension of the concept of convergence of crisp sequence. Proof: Let f be a crisp sequence in a metric space M. Then this can be considered as a fuzzy sequence A f. Now we have to prove that if the crisp sequence f converges then the fuzzy sequence A f converges at some level α> 0. Let f converge to l. Consider A f. Take α > 0. Let ε > 0 be given. Since the crisp sequence f converges to l, there exists n 0 є N such that d(x n, l) < ε for all n n 0. Now it is clear that A f (n,x) = 0 or 1 for all n є N, for all x є M. (i) (ii) For each n є N, we have x = x n where A f (n,x) = 1 α Now n n 0 and A f (n,x) α implies n n 0 and A f (n,x) = 1 which implies x = x n. Now d(x,l) = d(x n,l) < ε. Given ε > 0, there exists n 0 є N such that n n 0 and A f (n,x) α implies d(x,l) < ε. Hence A f converges to l. Hence the theorem. Result 3: In the previous theorem we have proved that for any crisp sequence f, f converges to l implies A f converges to l. Now we see that the converse is true. Theorem 4: Let f be a crisp sequence in a metric space M. If the corresponding fuzzy sequence A f converges to l at some level α> 0 then the crisp sequence f converges to l. Let f be a crisp sequence in metric space M. Let A f be the corresponding fuzzy sequence. Let A f converge to l at level α> 0. Let ε > 0 be given. Since A f converges to l there exists n 0 є N such that n n 0 and A f (n,x) α implies d(x,l) < ε. Now A f (n,x) α and α> 0 implies that A f (n,x) =1 which implies x = x n. Therefore we get d(x n,l ) < ε. Hence given ε > 0 there exists n 0 є N such that d(x n,l) < ε for all n n 0. Therefore the crisp sequence f converges to l.

6 704 M. Muthukumari, A. Nagarajan and M. Murugalingam Theorem 5: Let f be a crisp sequence in a metric space M. Then f converges to l if and only if the fuzzy sequence A f converges to l at some level α> 0. Follows from above two theorems. To get examples of convergent fuzzy sequences we prove some small results. Theorem 6: Let (a n ) and (b n ) be two crisp sequences in a metric space M converge to same limit l. Let A be the fuzzy sequence defined as A(n,x) = 1 if x = a n or x = b n and A (n,x) = 0 otherwise. Then A converges to l at any level α> 0. (a n ) converges to l and (b n ) converges to l. By definition of A, for each n є N,there exists a n є M such that A(n,a n ) = 1,Hence A(n,a n ) α Let ε > 0 be given. Since (a n ) converges to l, there exists n 1 є N such that d(a n,l ) < ε for all n n 1. Since (b n ) converges to l, there exists n 2 є N such that d(b n,l) < ε for all n n 2. Let n 0 = max {n 1,n 2 }. Now let n n 0 and A(n,x) α. Since α> 0, A(n,x) α implies that A(n,x) = 1 and hence x = a n or b n. Since n n 1, d(a n,l) < ε. Since n n 2, d(b n,l)< ε. Hence d(x,l) < ε. Therefore given ε < 0, there exists n 0 є N such that n n 0 and A(n,x) α implies d(x,l) < ε. Hence the fuzzy sequence A converges to l at any level α> 0. Result 4: Converse of the above theorem is true. Theorem 7: Let (a n ) and (b n ) be two crisp sequences in a metric space M. Let A be the fuzzy sequence defined as A(n,x) = 1 if x = a n or b n and A(n,x) = 0 otherwise. If A converges to l at some level α> 0 then (a n ) and (b n ) converge and converge to the same limit l. Proof: A(n,x) = 1 if x = a n or b n 0 otherwise

7 Fuzzy sequences in metric spaces 705 Fuzzy sequence A converges to l at some level α> 0. Claim : (a n ) converges to l. Let ε > 0 be given. Since A converges to l, there exists n 0 є N such that n n 0 and A(n,x) α implies d(x,l) < ε. Take n n 0. A(n,a n ) = 1 α. Hence d(a n,l) < ε. Hence given ε > 0, there exists n 0 є N such that d(a n, l ) < ε for all n n 0. Therefore (a n ) converges to l. Similarly (b n ) converges to l. Hence the theorem. Theorem 8: Let (a n ) and (b n ) be two crisp sequences in a metric space M. Let A be a fuzzy sequence in M defined as A(n,x) = 1 if x = a n or b n and A(n,x) = 0 otherwise. Then the fuzzy sequence A converges at some level α> 0 if and only if both (a n ) and (b n ) converge and they converge to the same limit. Follows from the previous theorems. Result 5: The above theorem can be extended to any finite number of crisp sequences. Theorem 9: Let { (a k n ) / k є K } be a collection of crisp sequences in a metric space M and K be any finite index set. Let A be a fuzzy sequence in M defined as A(n,x) = 1 if x = a k n for some k and A(n,x) = 0 otherwise. Then the fuzzy sequence A converges if and only if for each k, (a k n ) converges and all the sequences converge to the same limit in M. Similar to previous theorem. Theorem 10: Let α, β є (0,1] and α β. If a fuzzy sequence in a metric space M converges at level α then it converges at level β. Let M be a metric space and let A be a fuzzy sequence in M. α, β є [0,1] and α β. A converges to l at level α. Let ε > 0 be given. Since A converges to l at level α,

8 706 M. Muthukumari, A. Nagarajan and M. Murugalingam there exists n 0 є N, such that n n 0 and A(n,x) α implies d(x,l) <ε. Since α β, we have A(n,x) β implies A(n,x) α. Hence n n 0 and A(n,x) β implies n n 0 and A(n,x) α. Therefore d(x,l) < ε. Therefore given ε > 0, there exists n 0 є N such that n n 0 and A(n,x) β implies d(x,l) < ε. Hence A converges to l at level β. References 1. C.L. Chang, Fuzzy topological spaces, J.Math.Anal.Appl.24(1968), L. A. Zadeh, Fuzzy sets, Information and control, 8(1965) Received: February 23, 2014