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[{"key": "dc.contributor.advisor", "value": "Rajala, Kai", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Savolainen, Joonas", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-06-03T10:35:43Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-06-03T10:35:43Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/95447", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tutkitaan tasossa olevien joukkojen piirrett\u00e4vyytt\u00e4. Piirrett\u00e4vyys m\u00e4\u00e4ritell\u00e4\u00e4n kahden ty\u00f6kalun eli kyn\u00e4n ja kumin yhteisty\u00f6n\u00e4, jotka molemmat muodostavat 1-s\u00e4teisen kiekon kokoisen j\u00e4ljen tasoon. N\u00e4ist\u00e4 j\u00e4ljist\u00e4 kyn\u00e4n j\u00e4tt\u00e4m\u00e4 musta j\u00e4lki muodostaa tutkittavat piirrett\u00e4v\u00e4t joukot ja kumin j\u00e4tt\u00e4m\u00e4 valkoinen j\u00e4lki piirrett\u00e4vien joukkojen komplementin tason koskemattoman valkoisen alueen kanssa. Ty\u00f6kalun j\u00e4tt\u00e4m\u00e4n kiekon muotoisen j\u00e4ljen ollessa avoin puhutaan piirrett\u00e4vyydest\u00e4, mutta jos kiekko m\u00e4\u00e4ritell\u00e4\u00e4n suljetuksi, puhutaan SYK- eli suljetuin 1-s\u00e4teisten kiekoin piirrett\u00e4vyydest\u00e4.\nTutkielma koostuu kahdesta p\u00e4\u00e4tuloksesta, joista ensimm\u00e4inen on, ettei\n(2x2)-shakkiruudukko ole piirrett\u00e4v\u00e4. Kuitenkin py\u00f6rist\u00e4m\u00e4ll\u00e4 (2x2)-shakkiruudukon sis\u00e4kulmia joukosta saadaan paikallisesti piirrett\u00e4v\u00e4. Tutkittaessa (2x2)-shakkiruudukon piirrett\u00e4vyytt\u00e4 hy\u00f6dynnet\u00e4\u00e4n lemmaa, jonka avulla saadaan muodostettua ei-piirrett\u00e4vi\u00e4 joukkoja sek\u00e4 tutkittua my\u00f6s muiden joukkojen piirrett\u00e4vyytt\u00e4.\nTutkielman toisessa p\u00e4\u00e4lauseessa sanotaan, ett\u00e4 kaikki piirrett\u00e4v\u00e4t joukot ovat Lebesgue-mitallisia. Toisaalta on olemassa tason mitallisia osajoukkoja, jotka eiv\u00e4t ole piirrett\u00e4vi\u00e4. Lis\u00e4ksi tutkitaan tason piirrett\u00e4vien joukkojen kokoelman mahtavuutta verrattuna tason Lebesgue-mitallisten osajoukkojen kokoelmaan, tason konveksien joukkojen piirrett\u00e4vyytt\u00e4 sek\u00e4 piirrett\u00e4vien joukkojen kuuluvuutta Borel-joukkojen kokoelmaan.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2024-06-03T10:35:43Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-06-03T10:35:43Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "29", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "piirrett\u00e4vyys", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "mitallisuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "mahtavuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Joukkojen piirrett\u00e4vyys", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202406034211", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "joukot (matematiikka)", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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