- Email: [email protected]
Closed and convex fuzzy sets
Fuzzy Sets and Systems 110 (2000) 287–291 www.elsevier.com/locate/fss
Closed and convex fuzzy sets Yu-Ru Syau ∗ Department of Industrial Engineering, Yuan Ze University, Taoyuan Shian, 135 Yuan-Tung Road, Chung-Li City, Taiwan 320, ROC Received May 1997; received in revised form March 1998
Abstract In this paper, we establish a characterization theorem for closed fuzzy sets, and give two weak conditions that a closed fuzzy set is a convex fuzzy set. We also present some results for convex fuzzy sets, strictly convex fuzzy sets, and closed c 2000 Elsevier Science B.V. All rights reserved. fuzzy sets. Keywords: Membership functions; Various kinds of fuzzy sets; Convex fuzzy sets; Closed fuzzy sets
1. Introduction We adhere to the concepts and notations in [1] in which a fuzzy set : R n → [0; 1] was called convex if ( x1 + (1 − )x2 )¿(x1 ) + (1 − )(x2 ) for all x1 ; x2 ∈ supp() = {x: (x) ¿ 0}; ∈ [0; 1]; and strictly convex if strict inequality holds for all x1 ; x2 ∈ supp(); x1 6= x2 and ∈ (0; 1). A fuzzy set : R n → [0; 1] was called quasi-convex if ( x1 + (1 − )x2 )¿min{(x1 ); (x2 )} for all x1 , x2 ∈ supp(), ∈ [0; 1]; and strongly quasi-convex if strict inequality holds for all x1 , x2 ∈ supp(), x1 6= x2 and ∈ (0; 1). The concept of convexity is important for quantitative and qualitative studies in operations research. Yang [5] proposed the notion of closed fuzzy sets and obtained two weak conditions that a closed fuzzy set is a quasi-convex fuzzy set. In [1], Ammar and Metz ∗E-mail
address: [email protected] (Y.R. Syau)
de ned the fuzzy hypograph of a fuzzy set : R n → [0; 1], and proved that a fuzzy set is convex if and only if its fuzzy hypograph is convex. Motivated by Ammar and Metz [1], and Yang [5], we prove that a fuzzy set : R n → [0; 1] is closed if and only if its fuzzy hypograph is a closed subset of R n × (0; 1], and give two weak conditions that a closed fuzzy set is a convex fuzzy set. The two weak conditions greatly simplify criteria on convex fuzzy sets. We also present some results for convex fuzzy sets, strictly convex fuzzy sets, and closed fuzzy sets, which are motivated by Yang [6]. 2. Preliminaries Let R n denote the n-dimensional Euclidean space. De nition 2.1. The -level set of a fuzzy set : R n → [0; 1], ∈ [0; 1], denoted by [] , is de ned as [] = {x ∈ R n: (x)¿ }:
c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 0 8 2 - 7
Y.-R. Syau / Fuzzy Sets and Systems 110 (2000) 287–291
288
Remark 2.1. A fuzzy set : R n → [0; 1] is quasiconvex if and only if the -level set is a convex set for each ∈ [0; 1]. De nition 2.2 (Yang [5]). A fuzzy set : R n → [0; 1] is called closed if the -level set is a closed set for each ∈ [0; 1]. Remark 2.1. A fuzzy set : R n → [0; 1] is closed if and only if : R n → [0; 1] is upper semicontinuous. De nition 2.3 (Ammar and Metz [1]). The fuzzy hypograph of a fuzzy set : R n → [0; 1], denoted by f.hyp(), is de ned as n
f :hyp() = {(x; t): x ∈ R ; t ∈ (0; (x)]}: We recall Theorem 2.1 (Ammar and Metz [1, Theorem 1]). A fuzzy set : R n → [0; 1] is convex if and only if its fuzzy hypograph f :hyp() is a convex subset of R n × (0; 1]. 3. Main results The following result characterizes closed fuzzy sets : R n → [0; 1]. Theorem 3.1. A fuzzy set : R n → [0; 1] is closed if and only if its fuzzy hypograph f :hyp()={(x; t): x ∈ R n ; t ∈ (0; (x)]} is a closed subset of R n × (0; 1]. Proof. Suppose that : R n → [0; 1] is a closed fuzzy set. Hence : R n → [0; 1] is upper semicontinuous on R n . Let {(xn ; tn )} be a sequence of points of f.hyp() that converges to (x; t) in R n × (0; 1]. Since : R n → [0; 1] is upper semicontinuous and since (xn )¿tn , we must have (x) = lim (xn )¿ lim tn = t: n→∞
n→∞
It follows that (x; t) ∈ f :hyp(), and hence f :hyp() is a closed subset of R n × (0; 1]. Conversely, suppose that f :hyp() is a closed subset of R n × (0; 1]. For ∈ (0; 1], if [] has nite elements or is an empty set then it is obviously a closed subset of R n × (0; 1]. Assume that [] has in nitely many
elements, and consider a sequence of elements xn ∈ [] that converges to x. Since (xn )¿ , the sequence {(xn ; )} is a sequence of elements of f :hyp(), which converges to (x; ). Since f :hyp() is closed, (x; ) ∈ f :hyp(), that is, (x)¿ . It follows that x ∈ [] , and hence [] is closed. So, is a closed fuzzy set. This completes the proof. We establish the following, which provides a device for constructing closed and convex fuzzy sets on R n , as a corollary to Theorems 2.1 and 3.1. Theorem 3.2. Let H ⊂ R n × (0; 1]; and let (x) = sup{t: (x; t) ∈ H }: If H is a closed (resp. convex) subset of R n × (0; 1]; then : R n → [0; 1] is a closed (resp. convex) fuzzy set. Notice the usefulness here of the convention that a supremum over the empty set of (0,1] is 0. By using Theorem 2.1, Theorem 3.1 and the technique of Yang [5], one can prove the following result which gives weak conditions that a closed fuzzy set is a convex fuzzy set. Theorem 3.3. Let : R n → [0; 1] be a closed fuzzy set. Then is convex if and only if for all x1 ; x2 ∈ supp(); there exists a ∈(0; 1) ( depends on x1 ; x2 ); such that ( x1 + (1 − ) x2 )¿(x1 ) + (1 − )(x2 ): Proof. By Theorem 2.1, it is sucient to show that f :hyp() is a convex subset of R n × (0; 1]. By contradiction, suppose that there exist (x; t1 ), (y; t2 ) ∈ f :hyp(), and 0 ∈ [0; 1] such that q0 = 0 (x; t1 ) + (1 − 0 ) (y; t2 ) ∈= f :hyp(): Let A = f :hyp() ∩ I1
and
B = f :hyp() ∩ I2 ;
where I1 = { (x; t1 ) + (1 − )q0 : ∈ [0; 1]} and I2 = { q0 + (1 − ) (y; t2 ): ∈ [0; 1]}:
Y.-R. Syau / Fuzzy Sets and Systems 110 (2000) 287–291
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Since : R n → [0; 1] is closed, by Theorem 3.1, f :hyp() is a closed subset of R n × (0; 1]. Consequently, A and B are bounded and closed subset of R n × (0; 1], and q0 ∈= A, q0 ∈= B. Thus, there exist ˜ s) ∈ A and q2 = (y; ˜ t) ∈ B, with x, ˜ y˜ ∈ q1 = (x; supp() and s; t ∈ (0; 1], such that
set is a convex fuzzy set) but not convex. For example; consider the fuzzy set : R1 → [0; 1] de ned by ( 1=4 if x60; (x) = 1=2 if x ¿ 0:
min ka − q0 k = kq1 − q0 k
min kb − q0 k = kq2 − q0 k:
Proof. (a) is not a closed fuzzy set since []1=2 = (0; ∞) is not a closed subset of R1 . (b) To show that satis es the conditions of Theorem 3.3, it suces to show that for all x1 60, x2 ¿ 0, there exists a ∈(0; 1) ( depends on x1 , x2 ), such that
Hence, we have
( x1 + (1 − ) x2 )¿(x1 ) + (1 − )(x2 ):
f :hyp() ∩ (q1 ; q0 ] = ∅
Since for x1 60; x2 ¿ 0; there exists a ∈ (0; 1) such that x1 + (1 − ) x2 ¿ 0; it follows that
a∈A
and b∈B
and
( x1 + (1 − ) x2 ) =
f :hyp() ∩ [q0 ; q2 ) = ∅;
= 12 + 12 (1 − )
where (q1 ; q0 ] = { q1 + (1 − )q0 : ∈ (0; 1]}
¿ (x1 ) + (1 − )(x2 ):
and
(c) It is clear that is not a convex fuzzy set.
[q0 ; q2 ) = { q0 + (1 − )q2 : ∈ [0; 1)}:
Corollary 3.1. Let : R n → [0; 1] be a closed fuzzy set. Then is convex if and only if there exists a ∈ (0; 1) such that for all x1 ; x2 ∈ supp();
Therefore, f :hyp() ∩ (q1 ; q2 ) = ∅: On the other hand, by the hypothesis of the theorem, there exists a ∈ (0; 1) such that ( x1 + (1 − )x2 )¿(x1 ) + (1 − )(x2 ):
(3.1)
Since (x; ˜ s), (y; ˜ t) ∈ f :hyp(), we have (x)¿s ˜
1 2
and
(y)¿t: ˜
(3.2)
( x1 + (1 − ) x2 )¿(x1 ) + (1 − )(x2 ): Corollary 3.2. Let : R n → [0; 1] be a closed fuzzy set. Then is convex if and only if for all x1 ; x2 ∈ supp(); 12 x1 + 12 x2 ¿ 12 (x1 ) + 12 (x2 ):
That is,
Theorem 3.4. Let : R n → [0; 1] be a closed fuzzy set; and suppose that there exists a ∈ [0; 1] such that for all x; y ∈ R n ; u; v ∈ (0; 1) such that (x) ¿ u; (y) ¿ v;
q1 + (1 − )q2 = (x; ˜ s) + (1 − ) (y; ˜ t) ∈ f :hyp():
(x + (1 − )y) ¿ u + (1 − )v:
This contradicts that f :hyp()∩(q1 ; q2 ) = ∅: Thus, we proved that f :hyp() is a convex subset of R n × (0; 1].
Then : R n → [0; 1] is convex.
From inequalities (3.1) and (3.2), we obtain (x˜ + (1 − )y)¿s ˜ + (1 − )t:
Remark 3.1. We point out that there are fuzzy sets; which are not closed and satisfy the conditions of Theorem 3:3 (i.e.; the weak conditions that a closed fuzzy
Proof. By Theorem 3.3, it is sucient to show that for all x; y ∈ R n , there exists a ∈ (0; 1) such that ( x + (1 − )y)¿ (x) + (1 − )(y):
Y.-R. Syau / Fuzzy Sets and Systems 110 (2000) 287–291
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Suppose, on the contrary, there exist x1 ; x2 ∈ R n such that for all ∈ (0; 1), ( x1 + (1 − )x2 ) ¡ (x1 ) + (1 − )(x2 ):
(3.3)
For ¿ 0; since (x1 ) ¿ (x1 ) − , (x2 ) ¿ (x2 ) − , by the hypothesis of the theorem, we have
= (x1 ) + (1 − )(x2 ) − :
0¡
Since is an arbitrary positive real number, it follows that ( x1 + (1 − )x2 )¿ (x1 ) + (1 − )(x2 ); contradicting (3.3). It is clear that a strictly convex fuzzy set is a convex fuzzy set, but the converse is not true. Motivated by the work of Yang [6], we give some conditions which ensure that a convex fuzzy set : R n → [0; 1] is strictly convex. Theorem 3.5. Let : R n → [0; 1] be a convex fuzzy set. If there exists a ∈ (0; 1) such that for every pair of distinct points x1 ; x2 ∈ supp() ( x1 + (1 − )x2 ) ¿ (x1 ) + (1 − )(x2 ) (3.4) holds true; then is a strictly convex fuzzy set. Proof. Assume that is a convex fuzzy set which is not strictly convex. Then there exist x; y ∈ supp(), x 6= y, and ∈ (0; 1) such that (3.5)
From inequality (3.4) and the above equality, it is clear that 6= : (1) If 0 ¡ ¡ , let
z1 = x + 1 − y: (3.6)
z = x + (1 − )y
= x+ 1− y + (1 − )y = z1 + (1 − )y:
Since is a convex fuzzy set, from equality (3.6) and the above inequality, we obtain
contradicting (3.5). (2) If ¡ ¡ 1, that is
¿[(x1 ) − ] + (1 − )[(x2 ) − ]
Then,
(z) ¿ (z1 ) + (1 − )(y):
(z) ¿ (x) + (1 − )(y);
( x1 + (1 − )x2 )
(z) = (x) + (1 − )(y):
According to (3.4), we have
− ¡ 1; 1−
let z2 =
1−
− x+ y: 1− 1−
(3.7)
Thus, z = x + (1 − )y = x + (1 − )z2 : According to (3.4), we have (z) ¿ (x) + (1 − )(z2 ): Since is a convex fuzzy set, from equality (3.7) and the above inequality, we obtain (z) ¿ (x) + (1 − )(y); contradicting (3.5). This completes the proof. According to Theorems 3.3 and 3.5, we have the following corollary. Corollary 3.3. Let : R n → [0; 1] be a closed fuzzy set. If there exists a ∈ (0; 1) such that for every pair of distinct points x1 ; x2 ∈ supp() ( x1 + (1 − ) x2 ) ¿ (x1 ) + (1 − )(x2 ) holds true; then is a strictly convex fuzzy set. References [1] E. Ammar, J. Metz, On fuzzy convexity and parametric fuzzy optimization, Fuzzy Sets and Systems 49 (1992) 135–141. [2] J.-P. Aubin, Applied Abstract Analysis, Wiley-Interscience, New York, 1977.
Y.-R. Syau / Fuzzy Sets and Systems 110 (2000) 287–291 [3] R. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. [4] Y.R. Syau, On convex and concave fuzzy mappings, Fuzzy Sets and Systems 103 (1999) 163–168.
291
[5] X. Yang, A note on convex fuzzy sets, Fuzzy Sets and Systems 53 (1993) 117–118. [6] X. Yang, Some properties of convex fuzzy sets, Fuzzy Sets and Systems 72 (1995) 129–132.
Closed and convex fuzzy sets Yu-Ru Syau ∗ Department of Industrial Engineering, Yuan Ze University, Taoyuan Shian, 135 Yuan-Tung Road, Chung-Li City, Taiwan 320, ROC Received May 1997; received in revised form March 1998
Abstract In this paper, we establish a characterization theorem for closed fuzzy sets, and give two weak conditions that a closed fuzzy set is a convex fuzzy set. We also present some results for convex fuzzy sets, strictly convex fuzzy sets, and closed c 2000 Elsevier Science B.V. All rights reserved. fuzzy sets. Keywords: Membership functions; Various kinds of fuzzy sets; Convex fuzzy sets; Closed fuzzy sets
1. Introduction We adhere to the concepts and notations in [1] in which a fuzzy set : R n → [0; 1] was called convex if ( x1 + (1 − )x2 )¿(x1 ) + (1 − )(x2 ) for all x1 ; x2 ∈ supp() = {x: (x) ¿ 0}; ∈ [0; 1]; and strictly convex if strict inequality holds for all x1 ; x2 ∈ supp(); x1 6= x2 and ∈ (0; 1). A fuzzy set : R n → [0; 1] was called quasi-convex if ( x1 + (1 − )x2 )¿min{(x1 ); (x2 )} for all x1 , x2 ∈ supp(), ∈ [0; 1]; and strongly quasi-convex if strict inequality holds for all x1 , x2 ∈ supp(), x1 6= x2 and ∈ (0; 1). The concept of convexity is important for quantitative and qualitative studies in operations research. Yang [5] proposed the notion of closed fuzzy sets and obtained two weak conditions that a closed fuzzy set is a quasi-convex fuzzy set. In [1], Ammar and Metz ∗E-mail
address: [email protected] (Y.R. Syau)
de ned the fuzzy hypograph of a fuzzy set : R n → [0; 1], and proved that a fuzzy set is convex if and only if its fuzzy hypograph is convex. Motivated by Ammar and Metz [1], and Yang [5], we prove that a fuzzy set : R n → [0; 1] is closed if and only if its fuzzy hypograph is a closed subset of R n × (0; 1], and give two weak conditions that a closed fuzzy set is a convex fuzzy set. The two weak conditions greatly simplify criteria on convex fuzzy sets. We also present some results for convex fuzzy sets, strictly convex fuzzy sets, and closed fuzzy sets, which are motivated by Yang [6]. 2. Preliminaries Let R n denote the n-dimensional Euclidean space. De nition 2.1. The -level set of a fuzzy set : R n → [0; 1], ∈ [0; 1], denoted by [] , is de ned as [] = {x ∈ R n: (x)¿ }:
c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 0 8 2 - 7
Y.-R. Syau / Fuzzy Sets and Systems 110 (2000) 287–291
288
Remark 2.1. A fuzzy set : R n → [0; 1] is quasiconvex if and only if the -level set is a convex set for each ∈ [0; 1]. De nition 2.2 (Yang [5]). A fuzzy set : R n → [0; 1] is called closed if the -level set is a closed set for each ∈ [0; 1]. Remark 2.1. A fuzzy set : R n → [0; 1] is closed if and only if : R n → [0; 1] is upper semicontinuous. De nition 2.3 (Ammar and Metz [1]). The fuzzy hypograph of a fuzzy set : R n → [0; 1], denoted by f.hyp(), is de ned as n
f :hyp() = {(x; t): x ∈ R ; t ∈ (0; (x)]}: We recall Theorem 2.1 (Ammar and Metz [1, Theorem 1]). A fuzzy set : R n → [0; 1] is convex if and only if its fuzzy hypograph f :hyp() is a convex subset of R n × (0; 1]. 3. Main results The following result characterizes closed fuzzy sets : R n → [0; 1]. Theorem 3.1. A fuzzy set : R n → [0; 1] is closed if and only if its fuzzy hypograph f :hyp()={(x; t): x ∈ R n ; t ∈ (0; (x)]} is a closed subset of R n × (0; 1]. Proof. Suppose that : R n → [0; 1] is a closed fuzzy set. Hence : R n → [0; 1] is upper semicontinuous on R n . Let {(xn ; tn )} be a sequence of points of f.hyp() that converges to (x; t) in R n × (0; 1]. Since : R n → [0; 1] is upper semicontinuous and since (xn )¿tn , we must have (x) = lim (xn )¿ lim tn = t: n→∞
n→∞
It follows that (x; t) ∈ f :hyp(), and hence f :hyp() is a closed subset of R n × (0; 1]. Conversely, suppose that f :hyp() is a closed subset of R n × (0; 1]. For ∈ (0; 1], if [] has nite elements or is an empty set then it is obviously a closed subset of R n × (0; 1]. Assume that [] has in nitely many
elements, and consider a sequence of elements xn ∈ [] that converges to x. Since (xn )¿ , the sequence {(xn ; )} is a sequence of elements of f :hyp(), which converges to (x; ). Since f :hyp() is closed, (x; ) ∈ f :hyp(), that is, (x)¿ . It follows that x ∈ [] , and hence [] is closed. So, is a closed fuzzy set. This completes the proof. We establish the following, which provides a device for constructing closed and convex fuzzy sets on R n , as a corollary to Theorems 2.1 and 3.1. Theorem 3.2. Let H ⊂ R n × (0; 1]; and let (x) = sup{t: (x; t) ∈ H }: If H is a closed (resp. convex) subset of R n × (0; 1]; then : R n → [0; 1] is a closed (resp. convex) fuzzy set. Notice the usefulness here of the convention that a supremum over the empty set of (0,1] is 0. By using Theorem 2.1, Theorem 3.1 and the technique of Yang [5], one can prove the following result which gives weak conditions that a closed fuzzy set is a convex fuzzy set. Theorem 3.3. Let : R n → [0; 1] be a closed fuzzy set. Then is convex if and only if for all x1 ; x2 ∈ supp(); there exists a ∈(0; 1) ( depends on x1 ; x2 ); such that ( x1 + (1 − ) x2 )¿(x1 ) + (1 − )(x2 ): Proof. By Theorem 2.1, it is sucient to show that f :hyp() is a convex subset of R n × (0; 1]. By contradiction, suppose that there exist (x; t1 ), (y; t2 ) ∈ f :hyp(), and 0 ∈ [0; 1] such that q0 = 0 (x; t1 ) + (1 − 0 ) (y; t2 ) ∈= f :hyp(): Let A = f :hyp() ∩ I1
and
B = f :hyp() ∩ I2 ;
where I1 = { (x; t1 ) + (1 − )q0 : ∈ [0; 1]} and I2 = { q0 + (1 − ) (y; t2 ): ∈ [0; 1]}:
Y.-R. Syau / Fuzzy Sets and Systems 110 (2000) 287–291
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Since : R n → [0; 1] is closed, by Theorem 3.1, f :hyp() is a closed subset of R n × (0; 1]. Consequently, A and B are bounded and closed subset of R n × (0; 1], and q0 ∈= A, q0 ∈= B. Thus, there exist ˜ s) ∈ A and q2 = (y; ˜ t) ∈ B, with x, ˜ y˜ ∈ q1 = (x; supp() and s; t ∈ (0; 1], such that
set is a convex fuzzy set) but not convex. For example; consider the fuzzy set : R1 → [0; 1] de ned by ( 1=4 if x60; (x) = 1=2 if x ¿ 0:
min ka − q0 k = kq1 − q0 k
min kb − q0 k = kq2 − q0 k:
Proof. (a) is not a closed fuzzy set since []1=2 = (0; ∞) is not a closed subset of R1 . (b) To show that satis es the conditions of Theorem 3.3, it suces to show that for all x1 60, x2 ¿ 0, there exists a ∈(0; 1) ( depends on x1 , x2 ), such that
Hence, we have
( x1 + (1 − ) x2 )¿(x1 ) + (1 − )(x2 ):
f :hyp() ∩ (q1 ; q0 ] = ∅
Since for x1 60; x2 ¿ 0; there exists a ∈ (0; 1) such that x1 + (1 − ) x2 ¿ 0; it follows that
a∈A
and b∈B
and
( x1 + (1 − ) x2 ) =
f :hyp() ∩ [q0 ; q2 ) = ∅;
= 12 + 12 (1 − )
where (q1 ; q0 ] = { q1 + (1 − )q0 : ∈ (0; 1]}
¿ (x1 ) + (1 − )(x2 ):
and
(c) It is clear that is not a convex fuzzy set.
[q0 ; q2 ) = { q0 + (1 − )q2 : ∈ [0; 1)}:
Corollary 3.1. Let : R n → [0; 1] be a closed fuzzy set. Then is convex if and only if there exists a ∈ (0; 1) such that for all x1 ; x2 ∈ supp();
Therefore, f :hyp() ∩ (q1 ; q2 ) = ∅: On the other hand, by the hypothesis of the theorem, there exists a ∈ (0; 1) such that ( x1 + (1 − )x2 )¿(x1 ) + (1 − )(x2 ):
(3.1)
Since (x; ˜ s), (y; ˜ t) ∈ f :hyp(), we have (x)¿s ˜
1 2
and
(y)¿t: ˜
(3.2)
( x1 + (1 − ) x2 )¿(x1 ) + (1 − )(x2 ): Corollary 3.2. Let : R n → [0; 1] be a closed fuzzy set. Then is convex if and only if for all x1 ; x2 ∈ supp(); 12 x1 + 12 x2 ¿ 12 (x1 ) + 12 (x2 ):
That is,
Theorem 3.4. Let : R n → [0; 1] be a closed fuzzy set; and suppose that there exists a ∈ [0; 1] such that for all x; y ∈ R n ; u; v ∈ (0; 1) such that (x) ¿ u; (y) ¿ v;
q1 + (1 − )q2 = (x; ˜ s) + (1 − ) (y; ˜ t) ∈ f :hyp():
(x + (1 − )y) ¿ u + (1 − )v:
This contradicts that f :hyp()∩(q1 ; q2 ) = ∅: Thus, we proved that f :hyp() is a convex subset of R n × (0; 1].
Then : R n → [0; 1] is convex.
From inequalities (3.1) and (3.2), we obtain (x˜ + (1 − )y)¿s ˜ + (1 − )t:
Remark 3.1. We point out that there are fuzzy sets; which are not closed and satisfy the conditions of Theorem 3:3 (i.e.; the weak conditions that a closed fuzzy
Proof. By Theorem 3.3, it is sucient to show that for all x; y ∈ R n , there exists a ∈ (0; 1) such that ( x + (1 − )y)¿ (x) + (1 − )(y):
Y.-R. Syau / Fuzzy Sets and Systems 110 (2000) 287–291
290
Suppose, on the contrary, there exist x1 ; x2 ∈ R n such that for all ∈ (0; 1), ( x1 + (1 − )x2 ) ¡ (x1 ) + (1 − )(x2 ):
(3.3)
For ¿ 0; since (x1 ) ¿ (x1 ) − , (x2 ) ¿ (x2 ) − , by the hypothesis of the theorem, we have
= (x1 ) + (1 − )(x2 ) − :
0¡
Since is an arbitrary positive real number, it follows that ( x1 + (1 − )x2 )¿ (x1 ) + (1 − )(x2 ); contradicting (3.3). It is clear that a strictly convex fuzzy set is a convex fuzzy set, but the converse is not true. Motivated by the work of Yang [6], we give some conditions which ensure that a convex fuzzy set : R n → [0; 1] is strictly convex. Theorem 3.5. Let : R n → [0; 1] be a convex fuzzy set. If there exists a ∈ (0; 1) such that for every pair of distinct points x1 ; x2 ∈ supp() ( x1 + (1 − )x2 ) ¿ (x1 ) + (1 − )(x2 ) (3.4) holds true; then is a strictly convex fuzzy set. Proof. Assume that is a convex fuzzy set which is not strictly convex. Then there exist x; y ∈ supp(), x 6= y, and ∈ (0; 1) such that (3.5)
From inequality (3.4) and the above equality, it is clear that 6= : (1) If 0 ¡ ¡ , let
z1 = x + 1 − y: (3.6)
z = x + (1 − )y
= x+ 1− y + (1 − )y = z1 + (1 − )y:
Since is a convex fuzzy set, from equality (3.6) and the above inequality, we obtain
contradicting (3.5). (2) If ¡ ¡ 1, that is
¿[(x1 ) − ] + (1 − )[(x2 ) − ]
Then,
(z) ¿ (z1 ) + (1 − )(y):
(z) ¿ (x) + (1 − )(y);
( x1 + (1 − )x2 )
(z) = (x) + (1 − )(y):
According to (3.4), we have
− ¡ 1; 1−
let z2 =
1−
− x+ y: 1− 1−
(3.7)
Thus, z = x + (1 − )y = x + (1 − )z2 : According to (3.4), we have (z) ¿ (x) + (1 − )(z2 ): Since is a convex fuzzy set, from equality (3.7) and the above inequality, we obtain (z) ¿ (x) + (1 − )(y); contradicting (3.5). This completes the proof. According to Theorems 3.3 and 3.5, we have the following corollary. Corollary 3.3. Let : R n → [0; 1] be a closed fuzzy set. If there exists a ∈ (0; 1) such that for every pair of distinct points x1 ; x2 ∈ supp() ( x1 + (1 − ) x2 ) ¿ (x1 ) + (1 − )(x2 ) holds true; then is a strictly convex fuzzy set. References [1] E. Ammar, J. Metz, On fuzzy convexity and parametric fuzzy optimization, Fuzzy Sets and Systems 49 (1992) 135–141. [2] J.-P. Aubin, Applied Abstract Analysis, Wiley-Interscience, New York, 1977.
Y.-R. Syau / Fuzzy Sets and Systems 110 (2000) 287–291 [3] R. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. [4] Y.R. Syau, On convex and concave fuzzy mappings, Fuzzy Sets and Systems 103 (1999) 163–168.
291
[5] X. Yang, A note on convex fuzzy sets, Fuzzy Sets and Systems 53 (1993) 117–118. [6] X. Yang, Some properties of convex fuzzy sets, Fuzzy Sets and Systems 72 (1995) 129–132.