COMPOSITIONS OF FUZZY RELATIONS APPLIED TO VERYFICATION LEARNING OUTCOMES ON THE EXAMPLE OF THE MAJOR GEODESY AND CARTOGRAPHY

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1 Mreła A., Sokołov O. Compositios of fuzzy relatios applied to veryficatio learig outcomes o the example of the major Geodesy ad Cartography. Joural of Educatio, Health ad Sport. 205;55): ISSN DOI 0.528/zeodo Formerly Joural of Health Scieces. ISSN / X. Archives Deklaracja. Specyfika i zawartość merytorycza czasopisma ie ulega zmiaie. Zgodie z iformacją MNiSW z dia 2 czerwca 204 r., że w roku 204 ie będzie przeprowadzaa ocea czasopism aukowych; czasopismo o zmieioym tytule otrzymuje tyle samo puktów co a wykazie czasopism aukowych z dia 3 grudia 204 r. The joural has had 5 poits i Miistry of Sciece ad Higher Educatio of Polad parametric evaluatio. Part B item ). The Author s) 205; This article is published with ope access at Licesee Ope Joural Systems of Kazimierz Wielki Uiversity i Bydgoszcz, Polad ad Radom Uiversity i Radom, Polad Ope Access. This article is distributed uder the terms of the Creative Commos Attributio Nocommercial Licese which permits ay ocommercial use, distributio, ad reproductio i ay medium, provided the origial authors) ad source are credited. This is a ope access article licesed uder the terms of the Creative Commos Attributio No Commercial Licese which permits urestricted, o commercial use, distributio ad reproductio i ay medium, provided the work is properly cited. This is a ope access article licesed uder the terms of the Creative Commos Attributio No Commercial Licese which permits urestricted, o commercial use, distributio ad reproductio i ay medium, provided the work is properly cited. The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Received: Revised Accepted: COMPOSITIONS OF FUZZY RELATIONS APPLIED TO VERYFICATION LEARNING OUTCOMES ON THE EXAMPLE OF THE MAJOR GEODESY AND CARTOGRAPHY A. Mreła, O. Sokołov A. Mreła, Kujawsko-Pomorska Szkoła Wyższa w Bydgoszczy Polska, Bydgoszcz, ul. Toruńska a.mrela@kpsw.edu.pl O. Sokołov, Uiwersytet Mikołaja Koperika w Toruiu Polska, Toruń, ul Gagaria o.sokolov@umk.pl Abstract The paper presets discussio about usig mathematical fuctios i order to help academic teachers to verify acquiremet of learig outcomes by studets o the example of the major geodesy ad cartography. It is relatively easy to build fuzzy relatio describig levels of realizatio ad validatio learig outcomes durig subject examiatios ad the fuzzy relatio with studets grades is already built by teachers, the problem is to combie these two relatios to get oe which describes the level of acquirig learig outcomes by studets. There are two mai requiremets facig this combiatios ad the paper shows that the best combiatio accordig to these requiremets is algebraic compositio. Keywords: learig outcome, fuzzy relatio, algebraic compositio. Itroductio I 999 Polish Uder Secretary of State of Natioal Educatio siged the Bologa Declaratio ad bega the reform of educatio, icludig the higher educatio i order to create A Europe of Kowledge ad i the future the Europea Higher Educatio Area EHEA). The Bologa Declaratio of 9 Jue 999http://

2 Oe of importat elemets of the EHEA is the capacity of comparig educatioal achievemets of graduates of differet HEIs 2 i differet coutries. The solutio of this problem was the Europea Qualificatio Framework EQF), which is a traslatio tool that helps commuicatio ad compariso betwee qualificatios systems i Europe 3. I Polad, the Polish Qualificatio Framework was described accordig to the EQF. The Miistry of Sciece ad Higher Educatio issued a regulatio with the descriptio of the Polish Qualificatio Framework for Higher Educatio 4. Accordig to this regulatio all HEIs had to describe the set of learig outcomes for each major i the rage of kowledge, skills ad competeces. Matrices of learig outcomes Learig outcomes, which were described for each major, have to be acquired by studets ad properly validated by academic teachers. I tab., a few sample learig outcomes for the major geodesy ad cartography are preseted. Tab. Sample learig outcomes described for the major geodesy ad cartography Symbol Learig outcomes Kowledge K0 Graduate has kowledge of mathematics, physics ad chemistry used i formulatig ad solvig simple problems i the field of geodesy ad cartography K02 Graduate kows basic methods, techiques, tools ad materials used i solvig simple egieerig tasks i the field of geodesy ad cartography K03 Graduate has basic kowledge of kiematics, electromagetism, wave ad geometric optics, acoustics ad physics of the atmosphere K04 Graduate has a basic kowledge of the maagemet of the istitutio / compay, icludig quality maagemet Skills S0 Graduate plas ad carries out computer simulatios, iterprets the results ad draws coclusios S02 Graduate edits ad develops basic maps ad their derivatives S03 Graduate presets i the Polish laguage oral presetatio o specific issues i the field of geodesy ad cartography S04 Graduate obtais iformatio from the literature, databases ad other carefully selected sources i Polish ad Eglish; itegrates the obtaied iformatio, makes its iterpretatio, draws coclusios, formulates ad justifies opiios Competece C0 C02 Graduate uderstads the eed for learig lifelog Graduate is able to idetify properly the priorities for implemetig tasks specified by themselves or other people Source: the sample set of learig outcomes. After describig the set of learig outcomes for the major, academic teachers have to check accordace of this set with the set of learig outcomes described i the Regulatio 5. The ext step, which ca be doe by experts-academic teachers, is preparig the matrix which gives the level of realizatio ad validatio of each learig outcome while studyig each subject of the curriculum. Moreover, while preparig the matrix academic teachers check whether each subject is eeded for realizatio learig outcomes ad whether each learig outcome is properly taught by the set of subjects. 2 Higher Educatio Istitutios ROZPORZĄDZENIE MINISTRA NAUKI I SZKOLNICTWA WYŻSZEGO z dia 2 listopada 20 r. w sprawie Krajowych Ram Kwalifikacji dla Szkolictwa Wyższego D.U. z roku 20 Nr 253 Poz. 520). 5 Ibidem. 84

3 Mathematics MAT) Basic course i geodesy BCG) Geodesy ad satellite avigatiogsn) Adjustmet calculus ADC) Physics PH) Maagemet ad lad maagemet MLM) Costructio techiques CT) Geodesy ad iformatio techologygit) C-GEO Egieerig graphics EG) Field exercises FE) Diploma thesis DT) Lad iformatio systems LIS) Geographic iformatio systems GIS) Basics of earth ad soil sciece BESS) The good tool for checkig whether each subject is eeded for realizig the set of learig outcomes is the matrix which rows cotai learig outcomes ad colums subjects ad each elemet of this matrix shows the level of realizatio of the learig outcome durig studyig the subject comp. tab. 2). Tab. 2 Matrix describig the level of realizatio of learig outcomes while studyig subjects of the major geodesy ad cartography Subject Learig outcome K0.0 K K03 K S S S S04.0 C C Source: the sample matrix. After buildig the matrices which describe levels of realizatio ad validatio of the set of learig outcomes while studyig subjects from the curriculum geodesy ad cartography, the academic teachers have to cosider rules ad methods which let them establish whether studets have acquired each learig outcome ad whether acquirig of each of the learig outcomes is properly validated. For example, the teachers have to ask themselves whether they are able to positively aswer to this questio: whether studets who have got positive grades o the examiatio of Mathematics, has acquired the learig outcome K0 ad it has bee properly validated. Accordig to tab. 3, it ca be oticed that learig outcome K0 is validated by oly oe subject, Mathematics. It is more difficult to validate the acquirig of the learig outcome by a few subjects for example K02), especially i the case if studets get differet grades durig examiatios of differet subjects. Matrix of grades Academic teachers have to prepare the matrix with grades which studets get durig examiatios. All uiversities have described the gradig scale i their study regulatios ad 85

4 for most of the Polish uiversities the grades are positive itegers from to 5, from to 6 or from 2 to 5. Let us assume that values of the matrix of grades should belog to the iterval [0,], so there has to be defied a icreasig fuctio 6 that trasforms the studets grades to values belogig to the iterval [0,] comp. tab.3). Tab. 3 Grades of studets of the major geodesy ad cartography trasformed i such a way they belog to the iterval [0,] Studet Subject Paweł Nowak Igor Zabłocki Aa Nowik Barbara Siwak Piotr Tracki Zofia Pawlak Jolata Zobicka Reata Wartoś MAT BCG GSN AJ PH MLM CT GIT C-GEO EG FE DT LIS GIS BESS Source: the sample matrix. Fuzzy relatios While commuicatios, people use imprecise expressios, for example teachers say that they got good studets, studets were poorly prepared for the examiatio or studet X is very good at maths. Thus it is importat to fid out the way of speakig ad writig about achievemets of studets or about acquirig learig outcomes. Classical logic durig aswerig questios says yes or o ad uses umbers 0 or. For example, for the problem has studet X acquired learig effect Y?, classical logic ca oly say yes or o. Sice fuzzy logic uses imprecise expressios ad for the problem has the studet X acquired the learig effect Y?, fuzzy logic ca say studet X have acquired learig effect poorly, so the level of acquiremet of this effect is equal to 0.2. Let us quote the defiitios 7. Fuzzy set X, X ) is a pair of the set X ad the membership X : X [0, ], which for elemet x X describes the level of belogig of x to the set X. Moreover, fuzzy relatio R betwee sets X ad Y is the set of pairs x, y) X Y with the membership fuctio R : X Y [0, ]. Let R be a fuzzy relatio betwee learig outcomes ad subjects of the major geodesy ad cartography, which membership values are set i the tab. 3 ad deote the level of acquirig of these learig outcomes by these subjects. Let R2 be a fuzzy relatio betwee studets ad subjects which membership values are put to the tab. 4 ad deote the x, the f x ) f ). For example, the fuctio f x) 2x 3 6 A fuctio is icreasig if 2 icreasig. 7 L. Rutkowski, Metody i techiki sztuczej iteligecji, PWN, Warszawa, x x2 is 86

5 marks of these subjects which these studets have got durig examiatios trasformed to the iterval [0,]. Operatios of fuzzy relatios Sice the problem that is discussed i the paper is acquirig the learig outcomes by studets, these two matrices are ot adequate to solve this problem. There have to be defied the ext matrix-fuzzy relatio R 3 betwee learig outcomes ad studets which membership values describes the level of acquirig learig outcomes by studets. This fuzzy relatio could be defied as the compositio of fuzzy relatios R ad R 2. Let us quote some defiitios 8. Let be doe two fuzzy sets X, A), X, B) defied o the same set X. The product of these sets is a pair X, AB ) which membership fuctio is defied as follows: A B x) T A x), B x)), where a, a 2 X ad T is oe of the fuctios: T M a, a ) mi{ a, }, a, a ) max{ a a,0 }, T P a, a a a. 2 a2 T L 2 2 2) 2 The uio of these two sets is a pair X, AB ) which membership fuctio is defied as follows: A B x) S A x), B x)), where a, a 2 X ad S is oe of the fuctios: S M a, a ) max{ a, }, a, a ) mi{ a,}, a, a ) a a a. 2 a2 S L 2 a2 S P 2 2 a2 Let be doe two fuzzy relatios R x, y), x, y) ad P y, z), y, z) R P. The compositio type S T of fuzzy relatios R X Y ad P Y Z is a fuzzy relatios RP X Z with membership fuctio defied i the followig way: R P x, z) Sy Y T R x, y), P y, z))). Requiremets from the compositio of fuzzy relatios The compositio R3 of fuzzy relatios has to fulfill a few basic requiremets:. Values of the membership fuctio belog to the iterval [0,], so it is atural to accept that if the value of the membership fuctio of the fuzzy relatio R 3 betwee learig outcomes ad studets is equal to zero, amely R 3 LO_, studet _ m) 0, There ca be said that studet _ m did ot acquire the learig outcome LO_. 2. If the learig outcome is taught o the few subjects which are plaed i cosecutive semesters, there should be icrease i acquirig of this learig outcome, so values of the membership fuctio should be bigger. 8 Ibidem. 87

6 Compositio R 3 type max-mi I the case of the compositio type max-mi, fuctios T ad S are defied i the followig way: ) for two elemets a,a2 : T M a, mi{ a, a2}, S M a, max{ a, a2}, 2) for elemets a,,a : T a, a,..., a ) mi { a }, S M 2 i,2,..., i M a, a2,..., a) max i,2,..., { ai Whe these fuctios are used, the compositio of fuzzy relatios R ad R 2 tab.2 ad tab. 3), the fuzzy relatio R 3, is defied as follows: R3 K, P) max m,2,..., M{mi{ R K, m); R2 m, P)}} Usig this formula, values of the membership fuctios of R 3, ca be calculated. I tab. 4 there are preseted values of levels of acquirig learig outcome K02 by studets of the major geodesy ad cartography after studyig oe Basic course i geodesy), two Basic course i geodesy ad Adjustmet calculus) ad three subjects Basic course i geodesy, Adjustmet calculus ad Geodesy ad satellite avigatio). Tab. 4 Levels of acquirig learig outcome K02 by the studets of the major geodesy ad cartography with max-mi compositio after studyig oe, two ad three subjects Studets Paweł Igor Aa Barbara Piotr Zofia Jolata Reata Subjects Nowak Zabłocki Nowik Siwak Tracki Pawlak Zobicka Wartoś BCG BCG, ADC BCG, ADC, GSN Source: data calculated o the basis of data from tab. 3 ad tab. 3 Let us calculate the level of acquirig learig outcome K02 by studet Paweł Nowak. This learig outcome has bee taught o the three subjects: Basic course i geodesy, Adjustmet calculus ad Geodesy ad satellite avigatio. After studyig first subject Basic course i geodesy, the level of acquirig K02 is equal to }. max{mi{ 0.4;.0}} 0.4. After studyig two subjects: Basic course i geodesy ad Adjustmet calculus, the level of acquirig K02 is equal to max{mi{ 0.4;.0};mi{0.4;0.6}} 0.4, ad fially after studyig these three subjects, the level of acquirig K02 is equal to max{mi{ 0.4;.0};mi{0.4;0.6};mi{0.4;0.4};0;...;0}

7 Thus i this case there is o icrease i the level of acquirig learig outcome K0 by Paweł Nowak. Notice that oly i the case of studet Igor Zabłocki, there is a icrease i the level of acquirig learig outcome K0. Note that the secod requiremet is ot fulfilled i the case of the max-mi compositio. Summarizig, the max-mi compositio is ot appropriate for our purpose because i the case of a few subjects which realize ad validate the same learig outcome, there could be o icrease i the level of acquirig learig outcomes by studets. Thus geerally the secod requiremet is ot fulfilled. Compositio R 3 type Łukasiewicz I the case of the compositio type Łukasiewicz, fuctios T ad S are defied as follows: ) for two elemets a,a2 : T L a, max{ a a2,0}, S L a, mi{ a a2,}, 2) for elemets a,,a : T L a, a2,..., a) max a ),0, i i S L a, a2,..., a) mi ai,. i As i the case of max-mi compositio, after applyig Łukasiewicz type compositio, the membership fuctio is calculated usig the formula M R K, P) mi{ max{ R K, m) R m, P),0},}. 3 2 m The levels of acquirig learig outcome K02 by the studets of the major geodesy ad cartography after passig oe, two ad three subjects are put to the table 5. Tab. 5 Levels of acquirig learig outcome K02 by the studets of the major geodesy ad cartography with Łukasiewicz compositio after studyig oe, two ad three subjects Studets Paweł Igor Aa Barbara Piotr Zofia Jolata Reata Subjects Nowak Zabłocki Nowik Siwak Tracki Pawlak Zobicka Wartoś BCG BCG, ADC BCG, ADC, GSN Source: data calculated o the basis of data from tab. 2 ad tab. 3 Let us calculate the level of acquirig learig outcome K02 by studet Paweł Nowak. After passig first subject Basic course i geodesy, the level of acquirig K02 is equal to mi{max{ 0.4,0},} 0.4. After passig two subjects: Basic course i geodesy ad Adjustmet calculus, the level of acquirig K02 is equal to 89

8 mi{max{ 0.4,0} max{ ,0},} 0.4, ad fially after passig these three subjects, the level of acquirig K02 is equal to mi{max{ 0.4,0} max{ ,0} max{ ,0},} 0.4. Thus i this case there is o icrease i the level of acquirig learig outcome K02 by Paweł Nowak. Notice that i the case of studet Barbara Siwak, the level of acquirig K02 after studyig oe, two or eve three subject is still equal to 0. The levels of validatig K02 i the case of these three subjects were equal to positive umbers ad her grades were positive umbers as well, so the level of acquirig K02 by Barbara Siwak should ot be equal to zero. Summarizig, Łukasiewicz-type compositio R 3 of fuzzy relatios R ad R3 do ot fulfill the first ad secod requiremets. Compositio R 3 type algebraic I the case of the compositio type algebraic, fuctios T ad S are defied i the followig way: ) for two elemets a,a2 : T P a, aa 2, S P a, a a2 aa 2, 2) for elemets a,,a : T a, a,..., a ) a, P 2 i S a, a,..., a ) a ). P 2 i i Similarly like before, we costruct the compositio type algebraic, which is equal to M R3 K, P) R K, m) R2 m, P)), m calculate the values of relatio R3 for learig outcome K02 after passig these three subjects for each studets which are put them to the tab. 6. Tab. 6 Levels of acquirig learig outcome K02 by the studets of the major geodesy ad cartography with algebraic compositio after studyig oe, two ad three subjects Studets Paweł Igor Aa Barbara Piotr Zofia Jolata Reata Subjects Nowak Zabłocki Nowik Siwak Tracki Pawlak Zobicka Wartoś BCG BCG, ADC BCG, ADC, GSN Source: data calculated o the basis of data from tab. 2 ad tab. 3 Now, we calculate the level of acquirig learig effect K02 by studet Paweł Nowak. After passig the first subject, the level of acquirig K02 is equal to 0.4 ) 0.4. i 90

9 After passig two subjects, the level of acquirig learig effect K02 is equal to 0.4 ) ) Ad fially after passig these three subjects, the level of acquirig K02 is equal to 0.4) ) ) Notice that for all studets the level of acquirig learig outcome K02 is higher whe they have passed more subjects realizig this learig outcome). Aalyzig the data put i tab. 6, there ca be oticed that all values are positive, so if studets grades ad the level of validatig learig outcomes are positive the levels of acquirig learig outcomes are also positive. Theorem For each studet, the more subjects have bee validatig the learig outcome, the higher level of acquirig this learig outcome is. Proof. Let S deote the studet ad let K deote a learig outcome. Let r be a level of acquirig learig outcome K after studyig subjects, where,2,..., N ad N is the umber of subject which validate learig outcome K. Let us deote by a the level of validatig K by the subject ad let b be the grade of this subject. Assume that a, 0 for each,2,..., N. b If there is oe subject, the let r be a level of acquirig K, so r 0. Let N. Accordig to the defiitio of algebraic compositio: r ab ) a2b2 )... ab ). Let r a b ) a b )... a b ) a b ). 2 2 b b Sice a, b 0, a 0, so a. The a b ) a2b2 )... ab ) ab ) a2b2 )... ab ) a b ). Thus a b ) a2b2 )... ab ) ab ) a2b2 )... ab ) a b hece r r. Therefore, by the priciple of mathematical iductio, the more subjects have bee validatig the learig outcome, the higher level of acquirig this learig outcome is. I tab. 7 there are put levels of learig outcomes of the studets of the major geodesy ad cartography with the applicatio of algebraic compositio. ), 9

10 Tab. 7 Levels of acquirig learig outcome K02 by the studets of the major geodesy ad cartography with algebraic compositio Studets Paweł Igor Aa Barbara Piotr Zofia Jolata Reata Subjects Nowak Zabłocki Nowik Siwak Tracki Pawlak Zobicka Wartoś K K K K S S S S C C Source: data calculated o the basis of data from tab. 2 ad tab. 3 Notice that studets ca achieve the maximal value of acquirig learig outcomes, for example studet Aa Nowik achieved the level of acquirig learig outcome K03 equal to. Coclusios Nowadays it is required that teachers should verify acquirig learig outcomes by studets. Sice the umber of learig outcomes, which are described for each HEI major, belogs i most cases to the iterval 50, 80) ad their acquirig are validated o differet subjects, therefore this process is ot easy. Oe of the way to help teachers could be the computer system which calculates levels of acquirig learig outcomes by studets o the data give by teachers: levels of verifyig learig outcomes durig subjects examiatios ad studets grades. To achieve this goal, it is better to use fuzzy tha classical logic so fuzzy relatios are used to store data. The paper presets discussio which type of compositio of fuzzy relatios is the best oe for cosidered purpose accordig to two requiremets. Oly the algebraic compositio fulfills these two requiremets, amely if studets got positive grades, the levels of acquiremet of learig outcomes are positive ad the more subjects studets realize ad pass, the higher levels of acquiremet they achieve. LITERATURA. Rutkowski L., Metody i techiki sztuczej iteligecji, PWN, Warszawa, The Europea Higher Educatio Area i 202: Bologa Process Implemetatio Report. the Educatio. Audiovisual ad Culture Executive Agecy EACEA P9 Eurydice). April The Bologa Declaratio of 9 Jue eclaratio.pdf ROZPORZĄDZENIE MINISTRA NAUKI I SZKOLNICTWA WYŻSZEGO z dia 2 listopada 20 r. w sprawie Krajowych Ram Kwalifikacji dla Szkolictwa Wyższego D.U. z roku 20 Nr 253 Poz. 520). 92