{"id":924,"date":"2023-07-28T07:14:37","date_gmt":"2023-07-28T07:14:37","guid":{"rendered":"https:\/\/statorials.org\/pl\/ogol-operacji\/"},"modified":"2023-07-28T07:14:37","modified_gmt":"2023-07-28T07:14:37","slug":"ogol-operacji","status":"publish","type":"post","link":"https:\/\/statorials.org\/pl\/ogol-operacji\/","title":{"rendered":"Operacje na zbiorach: suma, przeci\u0119cie, uzupe\u0142nienie i r\u00f3\u017cnica"},"content":{"rendered":"

<\/p>\n

\n

Zbi\u00f3r<\/strong> to zbi\u00f3r element\u00f3w.<\/span><\/p>\n

Zbi\u00f3r oznaczamy du\u017c\u0105 liter\u0105, a elementy zbioru definiujemy za pomoc\u0105 nawias\u00f3w klamrowych. Za\u0142\u00f3\u017cmy na przyk\u0142ad, \u017ce mamy zbi\u00f3r o nazwie \u201eA\u201d z elementami 1, 2, 3. Zapisaliby\u015bmy to w nast\u0119puj\u0105cy spos\u00f3b:<\/span><\/p>\n

ZA = {1, 2, 3}<\/strong><\/span><\/p>\n

W tym samouczku wyja\u015bniono najcz\u0119\u015bciej u\u017cywane operacje na zbiorach<\/strong> w prawdopodobie\u0144stwie i statystyce.<\/span><\/p>\n

unia<\/strong><\/span> <\/h3>\n

\"Dzia\u0142anie<\/p>\n

Definicja:<\/strong> Suma<\/em> zbior\u00f3w A i B to zbi\u00f3r element\u00f3w wyst\u0119puj\u0105cych w A lub B.<\/span><\/p>\n

Ocena:<\/strong> A \u222a B<\/span><\/p>\n

Przyk\u0142ady:<\/strong><\/span><\/p>\n

    \n
  • {1, 2, 3} \u222a {4, 5, 6} = {1, 2, 3, 4, 5, 6}<\/span><\/li>\n
  • {1, 2} \u222a {1, 2} = {1, 2}<\/span><\/li>\n
  • {1, 2, 3} \u222a {3, 4} = {1, 2, 3, 4}<\/span><\/li>\n<\/ul>\n

    Skrzy\u017cowanie<\/strong><\/span> <\/h3>\n

    \"Operacja<\/p>\n

    Definicja:<\/strong> Przeci\u0119ciem<\/em> zbior\u00f3w A i B jest zbi\u00f3r element\u00f3w, kt\u00f3re znajduj\u0105 si\u0119 zar\u00f3wno w A, jak i B.<\/span><\/p>\n

    Notacja:<\/strong> A \u2229 B<\/span><\/p>\n

    Przyk\u0142ady:<\/strong><\/span><\/p>\n

      \n
    • {1, 2, 3} \u2229 {4, 5, 6} = {\u2205}<\/span><\/li>\n
    • {1, 2} \u2229 {1, 2} = {1, 2}<\/span><\/li>\n
    • {1, 2, 3} \u2229 {3, 4} = {3}<\/span><\/li>\n<\/ul>\n

      Komplement<\/strong> <\/h3>\n

      \"Uzupe\u0142niaj\u0105ce<\/p>\n

      Definicja:<\/strong> Dope\u0142nieniem<\/em> zbioru A jest zbi\u00f3r element\u00f3w, kt\u00f3re znajduj\u0105 si\u0119 w zbiorze uniwersalnym U, ale nie nale\u017c\u0105 do A.<\/span><\/p>\n

      Ocena:<\/strong> A’ lub Ac<\/sup><\/span><\/p>\n

      Przyk\u0142ady:<\/strong><\/span><\/p>\n

        \n
      • Je\u015bli U = {1, 2, 3, 4, 5, 6} i A = {1, 2}, to A c<\/sup> = {3, 4, 5, 6}<\/span><\/li>\n
      • Je\u015bli U = {1, 2, 3} i A = {1, 2}, to A c<\/sup> = {3}<\/span><\/li>\n<\/ul>\n

        R\u00f3\u017cnica<\/strong> <\/h3>\n

        \"Dzia\u0142anie<\/p>\n

        Definicja:<\/strong> R\u00f3\u017cnica<\/em> mi\u0119dzy zbiorami A i B to zbi\u00f3r element\u00f3w, kt\u00f3re znajduj\u0105 si\u0119 w A, ale nie w B.<\/span><\/p>\n

        Oceny:<\/strong> A\u2013B<\/span><\/p>\n

        Przyk\u0142ady:<\/strong><\/span><\/p>\n

          \n
        • {1, 2, 3} \u2013 {2, 3, 4} = {1}<\/span><\/li>\n
        • {1, 2} \u2013 {1, 2} = {\u2205}<\/span><\/li>\n
        • {1, 2, 3} \u2013 {4, 5} = {1, 2, 3}<\/span><\/li>\n<\/ul>\n

          R\u00f3\u017cnica symetryczna<\/strong> <\/h3>\n

          \"Symetryczna<\/p>\n

          Definicja:<\/strong> Symetryczna r\u00f3\u017cnica<\/em> zbior\u00f3w A i B to zbi\u00f3r element\u00f3w wyst\u0119puj\u0105cych w A lub B, ale nie w obu.<\/span><\/p>\n

          Ocena:<\/strong> A\u0394B<\/span><\/p>\n

          Przyk\u0142ady:<\/strong><\/span><\/p>\n

            \n
          • {1, 2, 3} \u0394 {2, 3, 4} = {1, 4}<\/span><\/li>\n
          • {1, 2} \u0394 {1, 2} = {\u2205}<\/span><\/li>\n
          • {1, 2, 3} \u0394 {4, 5} = {1, 2, 3, 4, 5}<\/span><\/li>\n<\/ul>\n

            Produkt kartezja\u0144ski<\/strong><\/span> <\/h3>\n

            \"Iloczyn<\/p>\n

            Definicja:<\/strong> Iloczyn kartezja\u0144ski<\/em> zbior\u00f3w A i B jest zbiorem uporz\u0105dkowanych par zbior\u00f3w A i B.<\/span><\/p>\n

            Ocena:<\/strong> A x B<\/span><\/p>\n

            Przyk\u0142ady:<\/strong><\/span><\/p>\n

              \n
            • Je\u015bli A = {H, T} i B = {1, 2, 3}, to A x B = {(H, 1), (H, 2), (H, 3), (T, 1), ( T, 2), (T, 3)}<\/span><\/li>\n
            • Je\u015bli A = {T, H} i B = {1, 2, 3}, to A x B = {(T, 1), (T, 2), (T, 3), (H, 1), ( H, 2), (H, 3)}<\/span><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

              Zbi\u00f3r to zbi\u00f3r element\u00f3w. Zbi\u00f3r oznaczamy du\u017c\u0105 liter\u0105, a elementy zbioru definiujemy za pomoc\u0105 nawias\u00f3w klamrowych. Za\u0142\u00f3\u017cmy na przyk\u0142ad, \u017ce mamy zbi\u00f3r o nazwie \u201eA\u201d z elementami 1, 2, 3. Zapisaliby\u015bmy to w nast\u0119puj\u0105cy spos\u00f3b: ZA = {1, 2, 3} W tym samouczku wyja\u015bniono najcz\u0119\u015bciej u\u017cywane operacje na zbiorach w prawdopodobie\u0144stwie i statystyce. unia Definicja: […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-924","post","type-post","status-publish","format-standard","hentry","category-przewodnik"],"yoast_head":"\nOperacje na zbiorach: suma, przeci\u0119cie, uzupe\u0142nienie i r\u00f3\u017cnica<\/title>\n<meta name=\"description\" content=\"Delikatne wprowadzenie do operacji na zestawach, \u0142\u0105cznie z definicjami i przyk\u0142adami.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/statorials.org\/pl\/ogol-operacji\/\" \/>\n<meta property=\"og:locale\" content=\"pl_PL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Operacje na zbiorach: suma, przeci\u0119cie, uzupe\u0142nienie i r\u00f3\u017cnica\" \/>\n<meta property=\"og:description\" content=\"Delikatne wprowadzenie do operacji na zestawach, \u0142\u0105cznie z definicjami i przyk\u0142adami.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/statorials.org\/pl\/ogol-operacji\/\" \/>\n<meta property=\"og:site_name\" content=\"Statorials\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-28T07:14:37+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/setops1.png\" \/>\n<meta name=\"author\" content=\"Benjamin Anderson\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Napisane przez\" \/>\n\t<meta name=\"twitter:data1\" content=\"Benjamin Anderson\" \/>\n\t<meta name=\"twitter:label2\" content=\"Szacowany czas czytania\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuta\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/statorials.org\/pl\/ogol-operacji\/\",\"url\":\"https:\/\/statorials.org\/pl\/ogol-operacji\/\",\"name\":\"Operacje na zbiorach: suma, przeci\u0119cie, uzupe\u0142nienie i r\u00f3\u017cnica\",\"isPartOf\":{\"@id\":\"https:\/\/statorials.org\/pl\/#website\"},\"datePublished\":\"2023-07-28T07:14:37+00:00\",\"dateModified\":\"2023-07-28T07:14:37+00:00\",\"author\":{\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\"},\"description\":\"Delikatne wprowadzenie do operacji na zestawach, \u0142\u0105cznie z definicjami i przyk\u0142adami.\",\"breadcrumb\":{\"@id\":\"https:\/\/statorials.org\/pl\/ogol-operacji\/#breadcrumb\"},\"inLanguage\":\"pl-PL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/statorials.org\/pl\/ogol-operacji\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/statorials.org\/pl\/ogol-operacji\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Dom\",\"item\":\"https:\/\/statorials.org\/pl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Operacje na zbiorach: suma, przeci\u0119cie, uzupe\u0142nienie i r\u00f3\u017cnica\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/statorials.org\/pl\/#website\",\"url\":\"https:\/\/statorials.org\/pl\/\",\"name\":\"Statorials\",\"description\":\"Tw\u00f3j przewodnik po kompetencjach statystycznych!\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/statorials.org\/pl\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pl-PL\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/6484727a4612df3e69f016c3129c6965\",\"name\":\"Benjamin Anderson\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pl-PL\",\"@id\":\"https:\/\/statorials.org\/pl\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/statorials.org\/pl\/wp-content\/uploads\/2023\/11\/Benjamin-Anderson-96x96.jpg\",\"contentUrl\":\"https:\/\/statorials.org\/pl\/wp-content\/uploads\/2023\/11\/Benjamin-Anderson-96x96.jpg\",\"caption\":\"Benjamin Anderson\"},\"description\":\"Cze\u015b\u0107, jestem Benjamin i jestem emerytowanym profesorem statystyki, kt\u00f3ry zosta\u0142 oddanym nauczycielem Statorials. 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