Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 560 https://internationalpubls.com Second Order Bipolar Fuzzy Matrix and its Application in Medical Diagnosis Muthamizhselvi. S *,1 , V. M. Vijayalakshmi 2 , M.Nila 3 , Vidhyapriya. P 4 *,1 Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education for Women, Coimbatore, Tamil Nadu - 601 043, muthamilselvi394@gmail.com 2 Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education for Women, Coimbatore, Tamil Nadu - 601 043, vmviji_sh@avinuty.ac.in 3 Department of Mathematics, PSGR Krishnammal College for Women, Coimbatore, Tamilnadu- 641004, brightmoon02@gmail.com 4 Department of Mathematics, PSGR Krishnammal College for Women, Coimbatore, Tamilnadu- 641004, pvidhyapriya19@gmail.com Article History: Received: 04-06-2024 Revised: 03-07-2024 Accepted: 30-07-2024 Abstract In this research article, firstly a new concept of second order bipolar fuzzy matrix (SOBPFM) is introduced and their basic properties and results are proved with examples. Secondly, the concept of symmetrical difference operator (SDO) over SOBPFM is introduced and various properties of SDO are discussed and verified over SOBPFM. Finally, an application on SOBPFM to develop a methodology in handling decision-making problem in medical field which enables clinicians to effectively assess and classify medical conditions, particularly in scenarios involving complex symptoms and overlapping disease manifestation. Keywords : second order bipolar fuzzy set (SOBPFS), second order bipolar fuzzy matrix (SOBPFM), symmetrical difference operator (SDO) 1. Introduction The notion of fuzzy set theory was introduced by Zadeh [14] in 1965, which is an universality of classical set theory. The utility of fuzzy set theory in decision making was first demonstrated by Bellman and Zadeh [2] in 1970. Since then many researchers have been working the process of dealing with decision making problems by applying fuzzy set theory. In 1975, Zadeh [15] proposed the concept of second order fuzzy set which is defined as a map from a set to . Using second order fuzzy sets Kalaichelvi [3] (2007) extended first order fuzzy topology of Chang and Lowen to second order fuzzy topology. The matrix is a fundamental concept in mathematics and computer science. It is widely used in various fields including physics, engineering, economics and computer graphics. The concept of fuzzy matrix was first defined by Thomsan [13] in 1977. As an extension of Boolean matrices, the theory of fuzzy matrices were developed by Kim and Roush [4]. In 2022, Saranya et al [12] studied the basic operations and properties of fuzzy matrices. A fuzzy matrix extends the concept of a traditional matrix by allowing entries to have degrees of membership in a set rather than precise value. In 1994, Zhang [16] proposed the concept of bipolar fuzzy set. Bipolar fuzzy set is an extension of the traditional fuzzy set theory allowing for the representation of uncertainty in both position and negative directions whose membership degree range in, - where the satisfactory degree of a certain property associated to the fuzzy set locates in the interval , - and the satisfactory degree of counter-property to the concerned fuzzy set locates in the interval, -. In 2024, Muthamizhselvi and Vijayalakshmi [6] proposed the concept of Second order bipolar fuzzy set and second order bipolar fuzzy topological space. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 561 https://internationalpubls.com In 2001, Kuratowski [5] defined new concept of symmetrical difference over ordinary sets through union, intersection and negation. In 2004, Anton Antonov [1] introduced SDO over IFM and shown that associativity was not true for IFM. In 2021, Muthuraji [10] analysed commutative monoid on symmetrical difference operator over IFM. In 2019, M. Pal and Sanjib Mondal [11] introduced bipolar fuzzy matrix. In 2023, Muthuraji and Anitha [7] proposed the recent technology using the bipolar fuzzy matrices with the aid of score function in agriculture. Muthuraji and Punitha Elizabeth [9] proposed application of bipolar fuzzy matrices in flood damage. Application of Bipolar Fuzzy Matrix in the Research of Crops on Agriculture was proposed by Muthuraji and Anitha [8] in 2023. In this paper, second order bipolar fuzzy set, second order bipolar fuzzy matrix and the concept of symmetrical difference operator (SDO) over SOBPFM are introduced. Also an application of second order bipolar fuzzy matrix is presented into a decision making problem and a general algorithm has been constructed to solve the problems in medical diagnosis. 2. Preliminaries 2.1 Definition Let be an arbitrary nonempty set. Let , -. A fuzzy set in is a map from into I. 2.2 Definition A second order fuzzy set on is a map ̂ where is the closed unit interval , -. 2.3 Definition Let be a nonempty set. Then a pair ( ) is called bipolar-valued fuzzy set or bipolar fuzzy set in , where , - and , -. The set of all bipolar fuzzy set in is denoted as BPF(X). 2.4 Definition A pair ̂ . ̂ ̂ / is called a second order bipolar fuzzy set in where ̂ , -, - such that ̂ ( )( ) , - and ̂ , -, - such that ̂ ( )( ) , -, where . 2.5 Definition Let be a matrix, [ ] , where [ ] , -, and , then is called fuzzy matrix. 2.6 Definition A bipolar fuzzy matrix .( ) / is defined as .( ) / .( ) / where ( ) ( ) , - and ( ) ( ) , - , where and denotes the rows and columns of the matrix. 2.7 Definition (Operations in bipolar fuzzy matrix) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 562 https://internationalpubls.com Let .( ) / .( ) / & .( ) / .( ) / be two bipolar fuzzy matrices. Define (i) .( ) ( ) / .( ) / where ( ) and is defined as ( ) { ( ) ( )} & ( ) { ( ) ( )}, , for every . (ii) .( ) ( ) / .( ) / where ( ) and is defined as ( ) { ( ) ( )} & ( ) { ( ) ( )}, , for every . (iii) ..( ) / / .(( ) ( ) ) / is defined as ( ) ( ) ( ), , for every . ( ) ( ) ( ), for every . 3. Second order bipolar fuzzy matrix 3.1 Definition Let be a nonempty set. A second order bipolar fuzzy matrix (SBPFM) .( ̂ ) / is defined as .( ̂ ) / (. ̂ ̂ / ) where . ̂ / ( )( ) , - and ( ̂ ) ( )( ) , - , , where . 3.2 Definition (Operations on second order bipolar fuzzy matrix) Let .( ̂ ) / (. ̂ ̂ / ) & .( ̂ ) / (. ̂ ̂ / ) be two second order bipolar fuzzy matrices. Define (iv) .( ̂ ) ( ̂ ) / .( ̂ ) / where ̂ . ̂ ̂ / and is defined as ̂ ( )( ) { ̂ ( )( ) ̂ ( )( )} & ̂ ( )( ) { ̂ ( )( ) ̂ ( )( )}, , for every for every I. (v) .( ̂ ) ( ̂ ) / .( ̂ ) / where ̂ . ̂ ̂ / and is defined as ̂ ( )( ) { ̂ ( )( ) ̂ ( )( )} & ̂ ( )( ) { ̂ ( )( ) ̂ ( )( )}, , for every for every I. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 563 https://internationalpubls.com (vi) ..( ̂ ) / / (.. ̂ / ( ̂ ) / ) is defined as . ̂ / ( )( ) ̂ ( )( ), , for every for every I. ( ̂ ) ( )( ) ̂ ( )( ), for every for every I. 3.3 Example Let ( ̂ ) ( ̂ ) be two second order bipolar fuzzy matrices. ( ̂ ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] ( ̂ ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] . Then compute (i) ( ̂ ) ( ̂ ) (ii) ( ̂ ) ( ̂ ) (iii) .( ̂ ) / . Solution: (i) ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] (ii) ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] (iii) .( ̂ ) / [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] . 3.4 Definition The transpose of second order bipolar fuzzy matrix .( ̂ ) / . ̂ ̂ / is defined as the second order bipolar fuzzy matrix .( ̂ ) / . ̂ ̂ / with .( ̂ ) / .( ̂ ) / for all and . The transpose of .( ̂ ) / is denoted as .( ̂ ) / . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 564 https://internationalpubls.com 3.5 Example .( ̂ ) / [ ( ) ( ) ( ) ] Then the transpose is given by .( ̂ ) / ,( ) ( ) ( )- 3.6 Definition A square matrix .( ̂ ) / . ̂ ̂ / with .( ̂ ) / { ( ) ( ) is called second order bipolar fuzzy identity matrix, denoted as ( ̂ ) . 3.7 Example ( ̂ ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] 3.8 Remark Let .( ̂ ) / . ̂ ̂ / be square matrix of order n and ̂ be second order bipolar fuzzy identity matrix. Then ( ̂ ) ̂ ̂ ( ̂ ) ( ̂ ) 3.9 Definition Let .( ̂ ) / . ̂ ̂ / be square matrix (order n). Then the trace of second order bipolar fuzzy matrix .( ̂ ) / is denoted by ( ̂ ) and is defined by ( ̂ ) ( . ̂ / ( ̂ ) ), where 3.10 Example ( ̂ ) [ ( ) ( ) ( ) ( ) ] Then the trace of SOBPFM .( ̂ ) / is given by ( ̂ ) ( . ̂ / ( ̂ ) ) ( * + * +) ( ) 3.11 Properties of second order bipolar fuzzy matrix Let ( ̂ ) ( ̂ ) and ( ̂ ) be three SOBPFM of order and respectively, then (i) ( ̂ ) .( ̂ ) ( ̂ ) / .( ̂ ) ( ̂ ) / ( ̂ ) (Associativity) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 565 https://internationalpubls.com (ii) ( ̂ ) .( ̂ ) ( ̂ ) / ( ̂ ) ( ̂ ) ( ̂ ) ( ̂ ) (Distributive law) The same results hold for second order bipolar fuzzy complement matrices (i) ( ̂ ) .( ̂ ) ( ̂ ) / .( ̂ ) ( ̂ ) / ( ̂ ) (Associativity) (ii) ( ̂ ) .( ̂ ) ( ̂ ) / ( ̂ ) ( ̂ ) ( ̂ ) ( ̂ ) (Distributive law) 3.12 Example ( ) ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) .( ̂ ) ( ̂ ) / [ ( ) ( ) ( ) ( ) ] ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) .( ̂ ) ( ̂ ) / [ ( ) ( ) ( ) ( ) ] Property (i) holds (ii) ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) .( ̂ ) ( ̂ ) / [ ( ) ( ) ( ) ( ) ] ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) ( ̂ ) ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ] Property (ii) holds Similarly we can prove for complement of SOBPFM. 3.13 Theorem Let ( ̂ ) and ( ̂ ) be two SOBPFMs (order n) and - scalar such that . Then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 566 https://internationalpubls.com (i) ( ̂ ̂ ) ( ̂ ) ( ̂ ) (ii) . ( ̂ ) / ( ̂ ) (iii) ( ̂ ) .( ̂ ) / Proof : (i) Let ( ̂ ) and ( ̂ ) be two SOBPFMs (order n) ( ̂ ) ( . ̂ / ( ̂ ) ) ( ̂ ) ( . ̂ / ( ̂ ) ) Then ( ̂ ̂ ) ( ̂ ) , where ̂ . ̂ ̂ /. By the definition of trace of SOBPFM, we have ( ̂ ) ( { {. ̂ / . ̂ / }} { {( ̂ ) ( ̂ ) }}) ( { . ̂ / . ̂ / } { ( ̂ ) ( ̂ ) }) ( ̂ ) ( ̂ ) (ii) . ( ̂ ) / ( ( . ̂ / ) . ( ̂ ) /) ( . ̂ / ( ̂ ) ) ( ̂ ) (iii) Proof is obvious. 4. Properties of SDO on SOBPFM In this section SDO denoted by is introduced over SOBPFM. Some properties are discussed. 4.1 Definition Let .( ̂ ) / and .( ̂ ) / are two SOBPFM of same order. The SDO over .( ̂ ) / and .( ̂ ) / using basic operation and complement is defined as Let .( ̂ ) / (. ̂ ̂ / ) & .( ̂ ) / (. ̂ ̂ / ) be .( ̂ ) / .( ̂ ) / (.( ̂ ̂ ) ( ̂ ̂ )/ ) .( ̂ ) / Where .( ̂ ) / (. ̂ ̂ / ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 567 https://internationalpubls.com ̂ ((. ̂ . ̂ / / .. ̂ / ̂ /) ) and ̂ (.( ̂ ( ̂ ) ) (( ̂ ) ̂ )/ ) 4.2 Lemma For any ( ̂ ) ( ̂ ) ( ) . Then (i) ( ̂ ) ( ̂ ) ( ̂ ) ( ̂ ) (ii) ( ̂ ) ( ̂ ) ( ̂ ) , where ( ̂ ) . ̂ ̂ / (iii) ( ̂ ) ( ̂ ) ( ̂ ) (iv) ( ̂ ) ( ̂ ) ( ̂ ) ( ̂ ) Proof: (i) From the above definition ( ̂ ) ( ̂ ) [(. ̂ . ̂ / / .. ̂ / ̂ / ( ̂ ( ̂ ) ) (( ̂ ) ̂ )) ] [(.. ̂ / ̂ / . ̂ . ̂ / / (( ̂ ) ̂ ) ( ̂ ( ̂ ) )) ] [(. ̂ . ̂ / / .. ̂ / ̂ / ( ̂ ( ̂ ) ) (( ̂ ) ̂ )) ] ( ̂ ) ( ̂ ) (ii) ( ̂ ) ( ̂ ) [(. ̂ ̂ / .. ̂ / ̂ / (( ̂ ) ̂ ) ( ̂ ̂ )) ] [. ̂ . ̂ / ( ̂ ) ̂ / ] .. ̂ / ( ̂ ) / (( ̂ ) ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 568 https://internationalpubls.com (iii) Proof is similar as (ii) (iv) ( ̂ ) ( ̂ ) [(. ̂ . ̂ / / .. ̂ / ̂ / ( ̂ ( ̂ ) ) (( ̂ ) ̂ )) ] [. ̂ . ̂ / ( ̂ ) ̂ / ] ( ̂ ( ̂ ) ) 4.3 Lemma For any ( ̂ ) ( ̂ ) ( ) .Then (( ̂ ) ) (( ̂ ) ) ( ̂ ) ( ̂ ) Proof : (( ̂ ) ) (( ̂ ) ) [(.. ̂ / ̂ / . ̂ . ̂ / / (( ̂ ) ̂ ) ( ̂ ( ̂ ) )) ] [(. ̂ . ̂ / / .. ̂ / ̂ / ( ̂ ( ̂ ) ) (( ̂ ) ̂ )) ] ( ̂ ) ( ̂ ) 4.4 Lemma For any ( ̂ ) ( ̂ ) and ( ̂ ) with same order. Then ( ̂ ) .( ̂ ) ( ̂ ) / .( ̂ ) ( ̂ ) / ( ̂ ) Proof: The above property is proved by an illustrative example as follows: ( ̂ ) [ ( ) ( ) ( ) ( ) ] (( ̂ ) ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) [ ( ) ( ) ( ) ( ) ] (( ̂ ) ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) [ ( ) ( ) ( ) ( ) ] (( ̂ ) ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 569 https://internationalpubls.com ( ̂ ) .( ̂ ) ( ̂ ) / [ ( ) ( ) ( ) ( ) ] ( ̂ ) ( ̂ ) [ ( ) ( ) ( ) ( ) ] ( ̂ ) .( ̂ ) ( ̂ ) / [ ( ) ( ) ( ) ( ) ] Hence ( ̂ ) .( ̂ ) ( ̂ ) / .( ̂ ) ( ̂ ) / ( ̂ ) . 5. An application on second order bipolar fuzzy matrix in medical diagnosis In this paper, we propose an algorithm to find what type of fever affects the different group of people the most. Let * + be the set of all fevers where Dengue, Swine flu, Malaria, Typhoid. Let * + be the set of all symptoms caused by the above types of fever where Decrease in platelet count, fatique, Lose of apetite, Eye pain. Let * + be the set of all patients of different age groups where Kids aging between 1-12 Teenagers aging between 13-19 Adult aging between 20-50 Old people aging between 50 & above Now three second order bipolar fuzzy matrices .( ̂ ) / which gives information about the types of fever for the common symptoms, .( ̂ ) / which indicates common symptoms that occur in the different age group of people and .( ̂ ) / which shows which common symptom indicates the intensity of the fever are constructed by a decision-maker. Then the event intensity relation .( ̂ ) / , conformability intensity relation .( ̂ ) / , the non-event intensity relation .( ̂ ) / and non-symptom intensity relation .( ̂ ) / are determined. 5.1 Algorithm Step 1: Enter the SOBPFM .( ̂ ) / which indicates an event of types of fever for the common symptoms. Step 2: Enter the SOBPFM .( ̂ ) / which indicates common symptoms that occur in the different age group of people. Step 3: Enter the second order bipolar fuzzy matrix .( ̂ ) / which indicates the confirmative relation given by .( ̂ ) / . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 570 https://internationalpubls.com Step 4: Compute the event intensity relation .( ̂ ) / .( ̂ ) / .( ̂ ) / . Step 5: Compute conformability intensity relation .( ̂ ) / .( ̂ ) / .( ̂ ) / . Step 6: Compute non-event intensity relation .( ̂ ) / .( ̂ ) / (.( ̂ ) / ) . Step 7: Compute non-symptom intensity relation .( ̂ ) / (.( ̂ ) / ) .( ̂ ) / . Step 8: Separate the positive and negative values in .( ̂ ) / 5.2 Problem Symptoms with fever: Symptoms Temperature Reference level Temperature level in patient Temperature level in [0,1] 101 0 -105 0 for 102.3 0 0.4 102 0 -106 0 for 101.2 0 0 99 0 -105 0 for 104.3 0 0.9 103 0 -104 0 for 103.8 0 0.8 Solution: Step 1: Matrix for event relation .( ̂ ) / shows the types of fever for the common symptoms                 )1.0,7.0()7.0,6.0()2.0,4.0()3.0,5.0( )4.0,8.0()4.0,3.0()8.0,1.0()2.0,4.0( )6.0,1.0()9.0,8.0()7.0,7.0()4.0,5.0( )9.0,2.0()5.0,9.0()8.0,6.0()6.0,7.0( 4 3 2 1 4321 s s s s ffff Step 2: The following SOBPFM .( ̂ ) / indicates the common symptoms that occurs in different age group of people.                 )8.0,9.0()5.0,5.0()3.0,6.0()2.0,9.0( )1.0,6.0()4.0,2.0()4.0,5.0()6.0,7.0( )6.0,2.0()2.0,1.0()1.0,3.0()8.0,6.0( )7.0,1.0()6.0,4.0()5.0,8.0()7.0,6.0( 4 3 2 1 4321 p p p p ssss Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 571 https://internationalpubls.com Step 3: Assuming that the matrix for confirmative relation .( ̂ ) /                 )3.0,6.0()4.0,7.0()8.0,9.0()2.0,5.0( )5.0,2.0()6.0,1.0()5.0,3.0()3.0,1.0( )7.0,8.0()3.0,1.0()1.0,5.0()2.0,4.0( )2.0,1.0()8.0,3.0()6.0,7.0()7.0,5.0( 4 3 2 1 4321 s s s s ffff The following four types of relations can be constructed using the relations .( ̂ ) / .( ̂ ) / and .( ̂ ) / . Step 4: The event intensity relation.( ̂ ) / .( ̂ ) / .( ̂ ) /                 )1.0,7.0()5.0,5.0()2.0,4.0()2.0,5.0( )1.0,6.0()4.0,2.0()4.0,1.0()2.0,4.0( )6.0,1.0()2.0,1.0()1.0,3.0()4.0,5.0( )7.0,1.0()5.0,4.0()5.0,6.0()6.0,6.0( 4 3 2 1 4321 p p p p ffff Step 5: The conformability intensity relation .( ̂ ) / .( ̂ ) / .( ̂ ) /                 )3.0,6.0()4.0,5.0()3.0,6.0()2.0,5.0( )1.0,2.0()4.0,1.0()4.0,3.0()3.0,1.0( )6.0,2.0()2.0,1.0()1.0,3.0()2.0,4.0( )2.0,1.0()6.0,3.0()5.0,7.0()7.0,5.0( 4 3 2 1 4321 p p p p ffff Step 6: The non-event intensity relation .( ̂ ) / .( ̂ ) / (.( ̂ ) / )      C ijbpA 0 ˆ                 )9.0,3.0()3.0,4.0()8.0,6.0()7.0,5.0( )6.0,2.0()6.0,7.0()2.0,9.0()8.0,6.0( )4.0,9.0()1.0,2.0()3.0,3.0()6.0,5.0( )1.0,8.0()5.0,1.0()2.0,4.0()4.0,3.0( 4 3 2 1 4321 s s s s ffff     3 ˆ ijbpA                 )8.0,3.0()3.0,4.0()3.0,6.0()2.0,5.0( )1.0,2.0()4.0,2.0()2.0,5.0()6.0,6.0( )4.0,2.0()1.0,1.0()1.0,3.0()6.0,5.0( )1.0,1.0()5.0,1.0()5.0,4.0()4.0,3.0( 4 3 2 1 4321 p p p p ffff Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 572 https://internationalpubls.com Step 7: Eventually, the non-symptom intensity relation .( ̂ ) / (.( ̂ ) / ) .( ̂ ) /      C Sijbp                 )2.0,1.0()5.0,5.0()7.0,4.0()8.0,1.0( )9.0,4.0()6.0,8.0()6.0,5.0()4.0,3.0( )4.0,8.0()8.0,9.0()9.0,7.0()2.0,4.0( )3.0,9.0()4.0,6.0()5.0,2.0()3.0,4.0( 4 3 2 1 4321 p p p p ssss                 )1.0,1.0()5.0,5.0()2.0,4.0()3.0,1.0( )4.0,4.0()4.0,3.0()6.0,1.0()2.0,3.0( )4.0,1.0()8.0,8.0()7.0,7.0()2.0,4.0( )3.0,2.0()4.0,6.0()5.0,2.0()3.0,4.0( 4 3 2 1 4321 p p p p ffff Step 8: Separate the positive and negative values in .( ̂ ) / (. ̂ / )             1.05.04.01.0 4.03.01.03.0 1.08.07.04.0 2.06.02.04.0 .( ̂ ) /                 1.05.02.03.0 4.04.06.02.0 4.08.07.02.0 3.04.05.03.0 6. Conclusion In this article, the definition of second order bipolar fuzzy matrix is presented and their properties are discussed. Also symmetrical difference operation has been extended over SOBPFM and its properties are studied. In addition to this, a decision making problem in the medical field has been presented where the resulting SOBPFM (. ̂ / ) concludes that the patient has the high intensity of swine flu and malaria and patient has low intensity of Dengue and typhoid. It is to be noted that intensity of fever in the different age group of people may vary depending upon the temperature levels recorded. 7. References [1] Antonov, A., Symmetrical Difference Over Intuitionistic Fuzzy Sets, Eighth Int. Conf. On IFSs, NIFS, 10(3), 33- 36. http://ifigenia.org/wiki/issue:nifs/10/3/33-36 [2] Bellman, R. E., Zadeh, L. A., Decision Making in a Fuzzy Environment, Management Sciences,1970, 17(4),141 – 164. https://doi.org/10.1287/mnsc.17.4.B141 [3] Kalaichelvi, A. (2007), Second Order Fuzzy Topological Spaces – I, ActaCienciaIndica, 33(3), 819-826.     4 ˆ ijbpA http://ifigenia.org/wiki/issue:nifs/10/3/33-36 https://doi.org/10.1287/mnsc.17.4.B141 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 573 https://internationalpubls.com [4] Kim, K. H., Roush, F. W., Generalized Fuzzy Matrices, Fuzzy Sets and Systems, 1980, 4, 293-315. https://doi.org/10.1016/0165-0114(80)90016-0 [5] Kuratowski, K., Introduction to Set Theory and Topology, PWN-Polish Scientific Publishers WARSZAWA, 2001 [6] Muthamizhselvi, S., Vijayalakshmi, V. M., Exploring Second Order Bipolar Fuzzy Topological Spaces for Enhanced Decision-Making in Medical Diagnosis, Journal of Harbin Engineering University, 2024, 45 (2), 94-103. [7] Muthuraji, T., Anitha, A., Recent Technology in Agriculture Using the Bipolar Fuzzy Matrices With the Aid of Score Function, Fifth IEEE International Conference on Electrical, Computer and Communication Technologies (ICECCT), 2023, 01-04. http://dx.doi.org/10.1109/ICECCT56650.2023.10179734 [8] Muthuraji, T., Anitha, A., Application of Bipolar Fuzzy Matrix in the Research of Crops on Agriculture, Indian Journal of Natural Science, 79, 59188- 59193. [9] Muthuraji. T., Punitha Elizabeth, P., “Application of Bipolar Fuzzy Matrices in Flood Damage”, Fifth IEEE International Conference on Electrical, Computer and Communication Technologies (ICECCT), 2023. https://doi.org/10.1109/ICECCT56650.2023.10179839. [10] Muthuraji, T., Commutative Monoid on Symmetrical Difference Operator Over Intuitionistic Fuzzy Matrices, Applications and Applied Mathematics, 2021, 16(1), 320-331. https://digitalcommons.pvamu.edu/aam/vol16/iss1/16. [11] Pal, M., Mondal, S., Bipolar Fuzzy Matrices, Soft Computing, 2019, 23(20), 9885-9897. [12] Saranya, G., Femila Mercy Rani, J., A Study on Basic Operations and Properties of Fuzzy Matrices and its Sections, International Journal of Arts, Science and Humanities, 2022, 10, 104-112. https://doi.org/10.34293/sijash. [13] Thomason, M. G., Convergence of Powers of a Fuzzy Matrix, J,mathAnal.Appl.,1977, 57, 476-480. https://doi.org/10.1016/0022-247X(77)90274-8 [14] Zadeh, L.A. (1965), Fuzzy Sets, Information and Control, 8(3), 338-353 https://doi.org/10.1016/S0019- 9958(65)90241-X [15] Zadeh, L.A. (1975), The Concept of a Linguistic Variable and its Application to Approximate Reasoning – I, Information Sciences, 8, 199 - 249. https://doi.org/10.1016/0020-0255(75)90036-5 [16] Zhang, W.R. (1994), Bipolar Fuzzy Sets and Relations : a Computational Framework for Cognitive Modelling and Multiagent Decision Analysis, Proc. of IEEE., 305-309. https://doi.org/10.1109/IJCF.1994.375115 . https://doi.org/10.1016/0165-0114(80)90016-0 http://dx.doi.org/10.1109/ICECCT56650.2023.10179734 https://doi.org/10.1109/ICECCT56650.2023.10179839 https://digitalcommons.pvamu.edu/aam/vol16/iss1/16 https://doi.org/10.34293/sijash https://doi.org/10.1016/0022-247X(77)90274-8 https://doi.org/10.1016/S0019-9958(65)90241-X https://doi.org/10.1016/S0019-9958(65)90241-X https://doi.org/10.1016/0020-0255(75)90036-5 https://doi.org/10.1109/IJCF.1994.375115