Hulgateoreetiliste avaldiste teisendamisega seotud ülesannete tüübid diskreetse matemaatika veebikeskkonna jaoks
Date
2012
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Tartu Ülikool
Abstract
Tartu Ülikooli aines „Diskreetse matemaatika elemendid“ kasutatakse mitmeid õpiprogramme, mille eesmärk on tudengite ja õppejõudude elu lihtsamaks teha. Seni ei ole kasutusel ühtegi hulgateooria ülesannete tüüpidega rakendust. Bakalaureusetöö eesmärgiks oli täiendada olemasolevat veebistöötavat lausearvutuse programmi hulgateooria ülesannetega. Selleks lisati rakendusele kaks hulgateooria teisendamisega seotud ülesannete tüüpi: samaväärsuse ja alamhulga seose tõestamine. Mõlema ülesandetüübi puhul on üldine lahenduskäik sarnane: kuvatakse kaks avaldist, mida saab üksteisest sõltumatult teisendada. Selleks peab lahendaja valima alamavaldise ja rakendama sellele mõnda eeldefineeritud reeglit. Kui avaldised on teisendatud sobivale kujule, tuleb esitada vastus. Alamhulga seose tõestamise ülesannetes tuleb seejärel veel põhjendada, miks seos kehtib. Lahenduse käigus tehtud vead kuvatakse kohe kasutajale ning tehtud vigade arv salvestatakse ka õppejõule. Viimane saab vaadata tudengi tulemusi ülesannete lõikes ning samuti konkreetseid lahendusi.
Töös kirjeldati õpiprogrammidest saadavat kasu, olemasolevaid programme, lisatud ülesandetüüpide hulgateoreetilist tausta ning nende implementatsiooni rakenduses. Samuti tutvustati uusi ülesande tüüpe õppejõudude ja tudengite vaatepunktist.
Loodud ülesannetetüübid on kasulikud nii tudengitele kui õppejõududele. Ülesannete läbi lahendamine aitab üliõpilasel materjali omandada. Lisaks lõppvastuse kontrollimisele annab programm ka vahepealset tagasisidet, näiteks kui üritatakse kasutada mõnda sobimatut teisendusreeglit või eksitakse alamavaldise valimisega. Õppejõud saavad programmi abil ülesandeid koostada ja tudengite tulemusi vaadata.
There are many propositional calculus programs used in the course of Elements of Discrete Mathematics in University of Tartu. The aim of these applications is to help students acquire practical skills and get instant feedback for their solutions. There were not any programs for set theory exercises, so the goal of the thesis is to add new exercise types to the existing discrete mathematics web application. The two added set theory exercise types were for proving equality of set theory expressions and subset relationship. For this, programmatic representation of set theory and propositional calculus operations were created. Also, many transformation rules were implemented to re-express formulae using different operations. The thesis explains the advantages of using programs in Elements of Discrete Mathematics class, gives short overview of set theory exercises and describes the implementation of added types and shows how they work in action. To implement new types, all of technologies of the existing systems were used. These include PHP and MySQL on the server-side and JavaScript, HTML and CSS on the client-side. PHP framework named DomFramework and jQuery, the JavaScript library, were used.
There are many propositional calculus programs used in the course of Elements of Discrete Mathematics in University of Tartu. The aim of these applications is to help students acquire practical skills and get instant feedback for their solutions. There were not any programs for set theory exercises, so the goal of the thesis is to add new exercise types to the existing discrete mathematics web application. The two added set theory exercise types were for proving equality of set theory expressions and subset relationship. For this, programmatic representation of set theory and propositional calculus operations were created. Also, many transformation rules were implemented to re-express formulae using different operations. The thesis explains the advantages of using programs in Elements of Discrete Mathematics class, gives short overview of set theory exercises and describes the implementation of added types and shows how they work in action. To implement new types, all of technologies of the existing systems were used. These include PHP and MySQL on the server-side and JavaScript, HTML and CSS on the client-side. PHP framework named DomFramework and jQuery, the JavaScript library, were used.