Akter, Afroja

Alim, Md. Abdul

Sultana, Abeda

English

Scitech Research Journals

                                                                                         Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218    Volume 4, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                   366|   SCITECH                                                                        Volume 4, Issue 3    RESEARCH ORGANISATION|                   Published online: July 11, 2015| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals  Supremum and Infimum Operations on Fuzzy Sets Mst. Afroja Akter1, Dr. Abeda Sultana2, Md. Abdul Alim3 1Faculty of Mathematics, CAAS, IUBAT-International University of Business Agriculture and Technology. 2Professor, Dept. of Mathematics, Jahanginagar University. 3Lecturer in Mathematics, BGMEA University of Fashion & Technology.  Abstract  In this paper we have shown comparison between classical set and fuzzy set. Also we have discussed some basic definitions and properties related to fuzzy set that is used in this paper. Then we collected three theorems by reviewing some papers, which were unproved there and we proved these theorems. We have reviewed some research papers with proper references to do our work.   Keywords: Fuzzy set; Membership function; Supremum; Infimum.  1. Introduction A well defined collection of objects or elements 𝑥 out of some universe 𝑋, is termed as set 𝐴 if and only if an element  𝑥  shows this property that either it belongs to the set 𝐴 and we can symbolically write 𝑥 ∈ 𝐴, or it is excluded, and we write 𝑥 ∉ 𝐴. This conception can be described by using a characteristic function 1𝐴  , which is a mapping of the form                              1𝐴:𝑋 → {0,1}  where,     1𝐴 𝑥 =  0; 𝑖𝑓 𝑥 ∉ 𝐴1; 𝑖𝑓 𝑥 ∈ 𝐴.  Against this background, fuzzy sets can be introduced as a generalization of conventional sets by allowing elements of a universe not only to entirely belong or not to belong to a specific set, but also to belong to the set to a certain grade. For the description of fuzzy sets, the characteristic function  1𝐴  of a crisp set 𝐴 can be generalized to a membership function  𝜇𝐴  for a fuzzy set [01] 𝐴, which is a mapping of the form 𝜇𝐴:𝑋 → [0,1] represents a fuzzy measure from a set-theoretical point of view. In general, a fuzzy set  𝐴 can thus be expressed by a set of pairs consisting of the elements  𝑥  of a universe 𝑋 and a certain grade of pre-assumed membership 𝜇𝐴(𝑥) of the form                                           𝐴 =   𝑥, 𝜇𝐴(𝑥)  𝑥 ∈ 𝑋, 𝜇𝐴 𝑥 ∈  0,1   Thus, a fuzzy set 𝐴 in 𝑋 is characterized by its membership function 𝜇𝐴:𝑋 → [0,1]                                                                                          Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218    Volume 4, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                   367|  and  𝜇𝐴(𝑥) is interpreted as the degree of membership of element  𝑥 in fuzzy set  𝐴 for each  𝑥 ∈ 𝑋 .  2. Some Basic Definitions and Operations 2.1. Definition [01]: Suppose 𝑓,𝑔 ∈ 𝐼𝑋  are fuzzy sets. Then the standard compliment of  𝑓 is another fuzzy set  𝑓:𝑋 → 𝐼; defined by,                             𝑓 𝑥 = 1 − 𝑓 𝑥 ,∀𝑥 ∈ 𝑋. 2.2. Definition [01]: Suppose  𝑓,𝑔 ∈ 𝐼𝑋   are two fuzzy sets. Then the standard union of  𝑓 and  𝑔  is also a fuzzy set in  𝐼𝑋   denoted by  𝑓 ∪ 𝑔 and is defined by,                     𝑓 ∪ 𝑔  𝑥 = max 𝑓 𝑥 ,𝑔 𝑥  = 𝑓 𝑥 ∨ 𝑔 𝑥 ;  ∀𝑥 ∈ 𝑋. 2.3. Definition [01]: Suppose 𝑓,𝑔 ∈ 𝐼𝑋  are two fuzzy sets. Then the standard intersection of  𝑓  and 𝑔  is also a fuzzy set in  𝐼𝑋  denoted by 𝑓 ∩ 𝑔 and is defined by,                           𝑓 ∩ 𝑔  𝑥 = min 𝑓 𝑥 ,𝑔 𝑥  = 𝑓 𝑥 ∧ 𝑔 𝑥 ;∀𝑥 ∈ 𝑋. 2.4. Definition [01]: Suppose  𝑓,𝑔 ∈ 𝐼𝑋  are fuzzy sets. Then we say that “𝒇 is a subset of 𝒈” denoted by “ 𝑓 ⊂𝑔”  iff  𝑓 𝑥 ≤ 𝑔 𝑥 ,∀𝑥 ∈ 𝑋. 2.5. Definition [05]: Let 𝑋 and 𝑌 be sets, and let 𝑓:𝑋 → 𝑌 be a function. For a fuzzy set 𝜇 in  𝑋, the image of  𝜇 under 𝑓 is the fuzzy set  𝑓(𝜇) in 𝑌 defined, for 𝑦 ∈ 𝑌, by the rule 𝑓 𝜇  𝑦 =  sup 𝜇 𝑧 : 𝑧 ∈ 𝑓−1 𝑦   𝑖𝑓 𝑓−1 𝑦 ≠ ∅0                                         𝑖𝑓 𝑓−1 𝑦 = ∅.  2.6. Definition [05]: Let 𝑋 and 𝑌 be sets, and let 𝑓:𝑋 → 𝑌 be a function. For a fuzzy set 𝜈 in  𝑌, the inverse image of  𝜈 under 𝑓 is the fuzzy set 𝑓−1(𝜈) in 𝑋 by the rule 𝑓−1 𝜈  𝑥 = 𝜈(𝑓 𝑥 )  for  𝑥 ∈ 𝑋.                                          (i.e,  𝑓−1 𝜈 = 𝜈 ∘ 𝑓). 3. Some Basic Theorems 3.1. Theorem [05]: For fuzzy sets  𝜇, 𝜇1 , 𝜇2 and  𝜈 in  𝑋: (i) 𝜈 ∩  𝜇1 ∪ 𝜇2 =  𝜐 ∩ 𝜇1 ∪  𝜐 ∩ 𝜇2  (ii) 𝜈 ∪  𝜇1 ∩ 𝜇2 =  𝜐 ∪ 𝜇1 ∩  𝜐 ∪ 𝜇2  (iii) 1 −  𝜇1 ∪ 𝜇2 =  1 − 𝜇1 ∩  1 − 𝜇2  (iv) 1 −  𝜇1 ∩ 𝜇2 =  1 − 𝜇1 ∪ (1 − 𝜇2). Proof: (i) ∀ 𝑥 ∈ 𝑋;               𝜈 ∩  𝜇1 ∪ 𝜇2   𝑥 = 𝜈 𝑥 ∧  𝜇1 ∪ 𝜇2 (𝑥)                           = 𝜈 𝑥 ∧ {𝜇1 𝑥 ∨ 𝜇2 𝑥 }                           =  𝜈 𝑥 ∧ 𝜇1 𝑥  ∨   𝜈 𝑥 ∧ 𝜇2 𝑥                             =   𝜈 ∩ 𝜇1  𝑥  ∨   𝜈 ∩ 𝜇2  𝑥                                                                                             Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218    Volume 4, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                   368|                             =   𝜈 ∩ 𝜇1 ∪  𝜈 ∩ 𝜇2  (𝑥)  ∴ 𝜈 ∩  𝜇1 ∪ 𝜇2 =  𝜐 ∩ 𝜇1 ∪  𝜐 ∩ 𝜇2 .   (ii)  𝜈 ∪  𝜇1 ∩ 𝜇2   𝑥 =  𝜈 𝑥 ∨  𝜇1 ∩ 𝜇2 (𝑥)                             = 𝜈 𝑥 ∨ {𝜇1 𝑥 ∧ 𝜇2 𝑥 }                             =  𝜈 𝑥 ∨ 𝜇1 𝑥  ∧   𝜈 𝑥 ∨ 𝜇2 𝑥                               =   𝜈 ∪ 𝜇1  𝑥  ∧   𝜈 ∪ 𝜇2  𝑥                               =   𝜈 ∪ 𝜇1 ∩  𝜈 ∪ 𝜇2  (𝑥).  ∴ 𝜈 ∪  𝜇1 ∩ 𝜇2 =  𝜐 ∪ 𝜇1 ∩  𝜐 ∪ 𝜇2 .  (iii)  1 −  𝜇1 ∪ 𝜇2   𝑥 = 1 𝑥 −  𝜇1 ∪ 𝜇2  𝑥                             = 1 −∨ {𝜇1 𝑥 , 𝜇2 𝑥 }                                                       = ∧  1 − {𝜇1 𝑥 ,𝜇2 𝑥 }  [By De Morgan’s law]                            =  1 − 𝜇1 𝑥  ∧  1 − 𝜇2 𝑥                                                               =  1 − 𝜇1  𝑥 ∧  1 − 𝜇2  𝑥                             =   1 − 𝜇1 ∩  1 − 𝜇2   𝑥     ∴ 1 −  𝜇1 ∪ 𝜇2 =  1 − 𝜇1 ∩  1 − 𝜇2 .  (iv)  1 −  𝜇1 ∩ 𝜇2   𝑥 = 1 𝑥 −  𝜇1 ∩ 𝜇2  𝑥                              = 1 −∧  𝜇1 𝑥 ,𝜇2 𝑥                               = ∨  1 −  𝜇1 𝑥 , 𝜇2 𝑥       [By De Morgan’s    law]                             =  1 − 𝜇1 𝑥  ∨  1 − 𝜇2 𝑥                               =  1 − 𝜇1  𝑥 ∨  1 − 𝜇2  𝑥                              =   1 − 𝜇1 ∪  1 − 𝜇2   𝑥     ∴ 1 −  𝜇1 ∩ 𝜇2 =  1 − 𝜇1 ∪  1 − 𝜇2 .  3.2. Theorem: For fuzzy sets  𝜇𝑖 𝑖 ∈ 𝐼 : (i) 𝜐 ∩ (   𝜇𝑖)𝑖∈𝐼 =   𝜐 ∩ 𝜇𝑖 𝑖∈𝐼 ,                                                                                          Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218    Volume 4, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                   369|   (ii) 𝜐 ∪   𝜇𝑖𝑖∈𝐼  =   𝜐 ∩ 𝜇𝑖 𝑖∈𝐼 ,  (iii) 1 − 𝜇𝑖𝑖∈𝐼 =   1 − 𝜇𝑖 𝑖∈𝐼 ,  (iv) 1 − 𝜇𝑖𝑖∈𝐼 =   1 − 𝜇𝑖 𝑖∈𝐼 . Proof: (i)  𝜐 ∩ ( 𝜇𝑖𝑖∈𝐼  ) 𝑥 =∧  𝜈 𝑥 , 𝜇𝑖 𝑥 𝑖∈𝐼                 =  (𝜈 𝑥 ∧ 𝜇𝑖 𝑥 )𝑖∈𝐼                 =  (𝜐 ∩ 𝜇𝑖) 𝑥  𝑖∈𝐼              ∴ 𝜐 ∩ ( 𝜇𝑖)𝑖∈𝐼 =   𝜐 ∩ 𝜇𝑖 𝑖∈𝐼   (ii)  𝜐 ∪ ( 𝜇𝑖)𝑖∈𝐼   𝑥 =∨  𝜈 𝑥 , 𝜇𝑖 𝑥 𝑖∈𝐼                  =  (𝜈 𝑥 ∨ 𝜇𝑖 𝑥 𝑖∈𝐼 )                =  (𝜐 ∪ 𝜇𝑖) 𝑥     𝑖∈𝐼                            ∴ 𝜐 ∪   𝜇𝑖𝑖∈𝐼  =   𝜐 ∪ 𝜇𝑖 𝑖∈𝐼 . (iii)     1 − 𝜇𝑖𝑖∈𝐼   𝑥 = 1 − 𝜇𝑖(𝑥)𝑖∈𝐼                 =  (1 − 𝜇𝑖)(𝑥)𝑖∈𝐼                   ∴ 1 − 𝜇𝑖𝑖∈𝐼 =   1 − 𝜇𝑖 𝑖∈𝐼 .   (iv)  1 − 𝜇𝑖𝑖∈𝐼   𝑥 = 1 − 𝜇𝑖𝑖∈𝐼  𝑥               =  (1 − 𝜇𝑖)(𝑥)𝑖∈𝐼                   ∴ 1 − 𝜇𝑖𝑖∈𝐼 =   1 − 𝜇𝑖 𝑖∈𝐼 . 3.3. Theorem [06] [07]: Let 𝑓 be a function from 𝑋 to 𝑌. If 𝛼 and 𝛼𝑖 , 𝑖 ∈ 𝐼, are fuzzy sets in 𝑋 and if  𝛽  and  𝛽𝑗 , 𝑗 ∈ 𝐼, are fuzzy sets in  𝑌, then the following relations are valid: (i) 𝑓 𝑓−1 𝛽  = 𝛽 when 𝑓 is onto 𝑌. (ii) 𝑓 ∩ 𝛼𝑖 ⊆ ∩ 𝑓 𝛼𝑖 . (iii) 𝑓−1 ∩ 𝛽𝑖 = ∩ 𝑓−1 𝛽𝑖 . (iv) 𝑓 ∪ 𝛼𝑖 = ∪ 𝑓 𝛼𝑖 . (v) 𝑓−1 ∪ 𝛽𝑖 = ∪ 𝑓−1 𝛽𝑖 . (vi) 𝑓 𝑓−1 𝛽 ∩ 𝛼 =  𝛽 ∩ 𝑓 𝛼 . Proof:  These results are all consequences of the definition of 𝑓 𝛼  and 𝑓−1(𝛽). (i) 𝑓 𝑓−1 𝛽   𝑦 = ∨  𝑓−1 𝛽  𝑥 :𝑓 𝑥 = 𝑦                                                     = ∨  𝛽 𝑓 𝑥  :𝑓 𝑥 = 𝑦 = 𝛽 𝑦                              ∴ 𝑓 𝑓−1 𝛽  = 𝛽 . (ii) 𝑓 ∩ 𝛼𝑖  𝑦 =  ∨𝑥  ∩𝑖   𝛼𝑖 𝑥 :𝑓 𝑥 = 𝑦                                                                                           Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218    Volume 4, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                   370|           ≤ ∩𝑖  ∨𝑥  𝛼𝑖 𝑥 :𝑓 𝑥 = 𝑦           = ∩ 𝑓 𝛼𝑖  𝑦 .                               ∴ 𝑓 ∩ 𝛼𝑖 ⊆ ∩ 𝑓 𝛼𝑖 .  (iii) 𝑓−1 ∩ 𝛽𝑖  𝑥 =  ∩ 𝛽𝑖 𝑓 𝑥  : 𝑥 ∈ 𝑋                                                   = ∩  𝛽𝑖 𝑓 𝑥  : 𝑥 ∈ 𝑋                                                  = ∩ 𝑓−1 𝛽𝑖  𝑥    ∴ 𝑓−1 ∩ 𝛽𝑖 = ∩ 𝑓−1 𝛽𝑖  .   (iv) 𝑓 ∪ 𝛼𝑖  𝑦 = ∨𝑥  ∪𝑖   𝛼𝑖 𝑥 :𝑓 𝑥 = 𝑦           = ∪𝑖∨𝑥   𝛼𝑖 𝑥 :𝑓 𝑥 = 𝑦           = ∪ 𝑓 𝛼𝑖  𝑦 .     ∴ 𝑓 ∪ 𝛼𝑖 = ∪ 𝑓 𝛼𝑖 .   (v)     𝑓−1 ∪ 𝛽𝑖  𝑥 =  ∪ 𝛽𝑖 𝑓 𝑥  : 𝑥 ∈ 𝑋    = ∪  𝛽𝑖 𝑓 𝑥  : 𝑥 ∈ 𝑋    = ∪ 𝑓−1 𝛽𝑖  𝑥                                    ∴ 𝑓−1 ∪ 𝛽𝑖 = ∪ 𝑓−1 𝛽𝑖 .  (vi)  𝑓 𝑓−1 𝛽 ∩ 𝛼  𝑦 = ∨   𝑓−1 𝛽 ∩ 𝛼  𝑥 :𝑓 𝑥 = 𝑦         = ∨  𝛽 𝑓 𝑥  ∩ 𝛼 𝑥 :𝑓 𝑥 = 𝑦         = 𝛽 𝑦 ∧  ∨  𝛼 𝑥 :𝑓 𝑥 = 𝑦          =  𝛽 ∩ 𝑓 𝛼   𝑦 .   ∴ 𝑓 𝑓−1 𝛽 ∩ 𝛼 =  𝛽 ∩ 𝑓 𝛼 . Conclusion Since its inception in 1965, as a generalization of classical set theory, fuzzy set has been applied to many mathematical areas such as algebra, graph theory, control theory, analysis, and operations research, topology and so on. In addition, it has been applied in practice in various disciplines such as control, data processing, engineering, management, stock market, medicine and so on.                                                                                          Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218    Volume 4, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                   371|    References [1] Zadeh L. A., Fuzzy sets, Information and Control, 8(1965) 338-353. [2] George J Klir, Yuan Bo, Fuzzy sets and fuzzy logic, Theory and Applications, Prentice-Hall Inc. N.J. U.S.A. 1995. [3] Lowen, Fuzzy set theorey, Department of Mathematics and Computer Science, University of Antwerp; Belgium, Basic Concepts, Techniques and Bibliography, Kluwer Academic Publishers Dordrecht/ Boston/ London. [4] Mitra N.K., Fuzzy set and fuzzy relational structure using T-norm and Matlab,  Department of Mathematics, University of Dhaka. [5] Steve Carlson, Fuzzy Sets and Fuzzy Topologies, Early ideals and Obstacles, Rose-Hulman Institute of Technology, Part I (May 10). [6] C.L. Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Application, 24,182-190(1968). [7] David H. Foster, Fuzzy Topological Groups, Journal of Mathematical Analysis and Application, Academic Press, New York and London, Vol. 67, No.2, February 1979. 

Supremum and Infimum Operations on Fuzzy Sets

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Supremum and Infimum Operations on Fuzzy Sets

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