{"id":72,"date":"2009-10-12T21:52:26","date_gmt":"2009-10-12T18:52:26","guid":{"rendered":"http:\/\/ramuuns.id.lv\/blog\/?p=72"},"modified":"2009-10-12T22:02:31","modified_gmt":"2009-10-12T19:02:31","slug":"kopu-teorijas-elementi","status":"publish","type":"post","link":"https:\/\/ramuuns.com\/blog\/2009\/10\/12\/kopu-teorijas-elementi\/","title":{"rendered":"Kopu teorijas elementi"},"content":{"rendered":"<p>Ieraksts rakstu s\u0113rij\u0101 lekciju pieraksti.<\/p>\n<p><span style=\"font-size:14pt\">1.1 Kopas j\u0113dziens<br \/>\n<\/span><\/p>\n<p>Def: kopa<\/p>\n<p>Ar v\u0101rdu kopa mat. saprot, t\u0101du j\u0113dzienu, kuram var viennoz\u012bm\u012bgi pateikt, ka k\u0101ds elements tai pieder vai nepieder<\/p>\n<p>Kopas parasti apz\u012bm\u0113 ar lielajiem lat\u012b\u0146u burtiem<\/p>\n<p>N &#8211; natur\u0101lie<\/p>\n<p>Z &#8211; veselie<\/p>\n<p>Q &#8211; racion\u0101lie<\/p>\n<p>R &#8211; re\u0101lie<\/p>\n<p>C &#8211; kompleksie<\/p>\n<p><span style=\"font-family:Symbol\">\uf04c\uf020<\/span>&#8211; tuk\u0161\u0101 kopa<span style=\"font-family:Times New Roman\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">Interv\u0101li:<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija1.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija2.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija3.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija4.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\">(a,b) &#8211; vektors &#8211; nevis interv\u0101ls<\/p>\n<p style=\"margin-left: 1pt\">3.kopas, kur<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija5.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija6.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija7.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija8.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 19pt\">Uzdo\u0161anas veidi<\/p>\n<ul>\n<li>Ar elementu sarakstu (uzskaitot visus elementus)<br \/>\n{a,b,c,d}<\/li>\n<li>\n<div>Ar rakst\u012br\u012bgo paz\u012bmi (liel\u0101m\/bezgal\u012bg\u0101m kop\u0101m)<\/div>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija9.gif\" alt=\"\" \/><\/li>\n<\/ul>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija10.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija11.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija12.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija13.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija14.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija15.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija16.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija17.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija18.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija19.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija20.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija21.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija22.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija23.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p>X &#8211; universs<\/p>\n<p>Hi &#8211; kopas A harakteristisk\u0101 funkcija<\/p>\n<p style=\"margin-left: 1pt\"><span style=\"font-size:14pt\">1.2 Matem\u0101tisk\u0101s lo\u0123ikas simboli<br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija24.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija25.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">! (paties\u012bb\u0101 izskat\u0101s, k\u0101 spogu\u013cots lielais krievu G) &#8211; neg\u0101cija<\/p>\n<p style=\"margin-left: 1pt\">&amp; &#8211; konjunkcija<\/p>\n<p style=\"margin-left: 1pt\">V &#8211; disjunkcija<\/p>\n<p style=\"margin-left: 1pt\">=&gt; implik\u0101cija<\/p>\n<p style=\"margin-left: 1pt\">&lt;=&gt; ekvivalence<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija26.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><span style=\"font-size:14pt\">1.3 Darb\u012bbas ar kop\u0101m<br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\ufffd<\/p>\n<p style=\"margin-left: 1pt\">A, B &#8211; kopas<\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija27.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija28.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija29.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija30.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija31.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija32.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija33.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija34.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija35.gif\" alt=\"\" \/><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija36.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><span style=\"font-size:14pt\">1.3.5 Oper\u0101ciju \u012bpa\u0161\u012bbas<br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\">Oper\u0101cijas tiks veiktas, ar oper\u0101cij\u0101m A,B,C, kuras ir univers\u0101lkopas X apak\u0161kopas1.<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija37.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija38.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">2.<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija39.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">3.<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija40.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">4.<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija41.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija42.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija43.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">5.<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija44.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija45.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">6.<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija46.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija47.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">\n<p><span style=\"font-size:14pt\">1.3.6 Kopas papildin\u0101jums<br \/>\n<\/span><\/p>\n<p>Kopas A papildin\u0101jums ir tie un tikai tie elementi, kas nepieder kopai A. (papildin\u0101jumam b\u016btisks ir Universs, t.i. Kopa pret kuru tiek veikts papildin\u0101jums)<\/p>\n<p style=\"margin-left: 1pt\"><span style=\"font-size:14pt\">1.4 Kopu saimes<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija48.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija49.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija50.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><span style=\"font-size:14pt\">1.5 Dekarta reizin\u0101jums<br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\">1.5.1. Korte\u017ea j\u0113dziens<\/p>\n<p style=\"margin-left: 1pt\">Korte\u017es &#8211; sak\u0101rtota elementu virkn\u012bte (a,b,..)<br \/>\n(a,b) != {a,b} &#8211; iek\u0161 {a,b} sec\u012bba nav svar\u012bga, savuk\u0101rt iek\u0161 korte\u017ea &#8211; sec\u012bba ir svar\u012bga.<br \/>\nKorte\u017eu piem\u0113ri &#8211; Vektors, Matricas rinda, matricas kolonna.<\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\">1.5.2. Dekarta reizin\u0101jums<\/p>\n<p style=\"margin-left: 1pt\">A,B &#8211; kopas<\/p>\n<p style=\"margin-left: 1pt\">AxB = kopu A un B dekarta reizin\u0101jums<\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija51.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">1.5.3. Dekarta reizin\u0101juma \u012bpa\u0161\u012bbas<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija52.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija53.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija54.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija55.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija56.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija57.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija58.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija59.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><span style=\"font-size:14pt\">1.5 Funkcijas<\/span><\/p>\n<p style=\"margin-left: 1pt\">1.6.1. Funkcijas visp\u0101r\u012bga def.<\/p>\n<p style=\"margin-left: 1pt\">X,Y &#8211; kopas, f &#8211; funkcija, Df &#8211; funkcijas defin\u012bcijas kopa<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija60.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">f ir likums, saska\u0146\u0101 ar kuru, katram funkcijas defin\u012bcijas kopas Df elementam x ir piek\u0101rtots viens kopas Y elements y, kuru apz\u012bm\u0113 ar f(x) un sauc par funkcijas v\u0113rt\u012bbu punkt\u0101 x<\/p>\n<p style=\"margin-left: 1pt\">X &#8211; funkcijas f starta kopa<\/p>\n<p style=\"margin-left: 1pt\">Y &#8211; funkcijas f fini\u0161a kopa<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija61.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\">f- var uzdot ar tabulu (ja X &amp; Y ir gal\u012bgas kopas), grafiku vai anal\u012btiski (f(x) = x2)<\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\">Grafiks ir kopa:<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija62.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija63.gif\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p>f(x) = x2<\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija64.png\" alt=\"\" \/><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija65.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija66.gif\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 17pt\">1.6.2. Funkciju klasifik\u0101cija<\/p>\n<p style=\"margin-left: 17pt\">\n<ul>\n<li>Visur defin\u0113a funkcija &lt;==&gt; D<sub>f<\/sub> =X<span style=\"font-family:Arial\"><br \/>\n<\/span><\/li>\n<li>Sirjekcija &lt;==&gt; R<sub>f<\/sub> = Y<span style=\"font-family:Arial\"><br \/>\n<\/span><\/li>\n<li>Injekcija &lt;==&gt; visiem y no R<sub>f<\/sub> eksist\u0113 viens vien\u012bgs x no D<sub>f<\/sub>, t\u0101ds, ka y=f(x)<span style=\"font-family:Arial\"><br \/>\n<\/span><\/li>\n<li>Bijekcija &lt;==&gt; visur defin\u0113ta funkcija &amp; injekcija &amp; sirjekcija<\/li>\n<li>Ja X = R, Y &#8211; R, &#8211; viena argumenta (re\u0101las) funkcijas<\/li>\n<li>\n<div>Ja X = R<sup>n<\/sup>, Y -R, &#8211; vair\u0101ku argumentu (re\u0101lu) funkcijas<br \/>\nJa X = R<sup>n<\/sup>, Y -R<sup>k<\/sup> &#8211; vair\u0101ku argumentu vektorfunkcijas<span style=\"font-family:Arial\"><br \/>\n<\/span><\/div>\n<\/li>\n<li>\n<div>Ja X = C, Y &#8211; C, &#8211; kompleks\u0101 main\u012bg\u0101 funkcijas<span style=\"font-family:Arial\"><br \/>\n<\/span><\/div>\n<p>Ja Df = N &#8211; tad t\u0101 ir skait\u013cu virkne (a1,a2, &#8230;, an) &#8211; re\u0101la viena argumenta funkciju speci\u0101lgad\u012bjums<\/li>\n<\/ul>\n<p style=\"margin-left: 23pt\">Pamatelement\u0101r\u0101s funkcijas<\/p>\n<p style=\"margin-left: 23pt\">\n<ol>\n<li>Konstantes<br \/>\nf(x) = C, kur C = ir re\u0101ls skaitlis &#8211; Df = R<\/li>\n<li>Pak\u0101pes f-jas<br \/>\nf(x)=x<sup>r<\/sup> Df &#8211; atkar\u012bgs no r &#8211; ja r = 2 &#8211; Df = R, r = 1\/2 &#8211; Df = R \\ {0},<\/li>\n<li>Eksponentfunkcijas<br \/>\nf(x) = a<sup>x<\/sup><br \/>\na!= 1 &amp; a&gt;0<\/li>\n<li>Logaritmisk\u0101s f-jas<br \/>\nf(x)=log<sub>a<\/sub>(x)<br \/>\na!= 1 &amp; a&gt;0<\/li>\n<li>Trigonometrisk\u0101s funkcijas<br \/>\nf(x) = sin(x)| cos(x)| tg(x)| ctg(x)<\/li>\n<li>Invers\u0101s trigonometrisk\u0101s funkcijas<br \/>\nf(x) = arcSin(x)|arcCos(x)|arcTg(x)|arcCtg(x)<\/li>\n<li>\n<div>Ciklometrisk\u0101s funkcijas<br \/>\nf(x) = sinh(x)|cosh(x)|tanh(x)|ctanh(x)<\/div>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija67.gif\" alt=\"\" \/><br \/>\nFunkciju var ieg\u016bt no pamatelement\u0101r\u0101m funkcij\u0101m izmantojot gal\u012bg\u0101 skait\u0101 oper\u0101cijas + &#8211; * \/ kompoz\u012bcija<\/li>\n<\/ol>\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija68.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">1.6.3. Viena re\u0101la argumenta funkciju pamat\u012bpa\u0161\u012bbas<\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija69.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija70.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija71.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija72.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\">f- monotona &lt;=&gt; (f &#8211; nedilst) V (f &#8211; nedilst)<\/p>\n<p style=\"margin-left: 1pt\">f &#8211; st. monotona &lt;=&gt; (f &#8211; aug) V (f -dilst)<\/p>\n<p style=\"margin-left: 1pt\">f- ierobe\u017eota no aug\u0161as &lt;=&gt; <img loading=\"lazy\" class=\"alignnone\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%5Cexists%20M%20%5Cin%20%5Cmathbb%7BR%7D%20%3A%20%5Cforall%20x%20%5Cin%20D_f%20f(x)%20%5Cle%20M\" alt=\"\" width=\"201\" height=\"19\" \/><\/p>\n<p style=\"margin-left: 1pt\">f- ierobe\u017eota no apak\u0161as  &lt;=&gt; <img loading=\"lazy\" class=\"alignnone\" src=\"http:\/\/chart.apis.google.com\/chart?cht=tx&amp;chs=1x0&amp;chf=bg,s,FFFFFF00&amp;chco=000000&amp;chl=%5Cexists%20M%20%5Cin%20%5Cmathbb%7BR%7D%20%3A%20%5Cforall%20x%20%5Cin%20D_f%20f(x)%20%5Cge%20M\" alt=\"\" width=\"201\" height=\"19\" \/><\/p>\n<p style=\"margin-left: 1pt\">f &#8211; ierobe\u017eota &lt;=&gt; f &#8211; ierobe\u017eota no aug\u0161as &amp; f -ierobe\u017eota no apak\u0161as<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija73.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija74.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija75.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">1.6.4. Saliktas funkcijas j\u0113dziens<\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\">Piem\u0113ri:<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija76.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">1.6.5. Invers\u0101s funkcijas j\u0113dziens<\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\">X,Y &#8211; kopas<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija77.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija78.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\">\n<p><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija79.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 1pt\">1.6.6. Att\u0113li un pirmt\u0113li<\/p>\n<p style=\"margin-left: 1pt\">f(A) &#8211; kopas att\u0113ls<\/p>\n<p style=\"margin-left: 1pt\">f-1(B) &#8211; kopas B pirmt\u0113ls<\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija80.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija81.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija82.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 1pt\"><span style=\"font-size: 19px;\">1.7. Kopas apjoms<\/span><\/p>\n<p style=\"margin-left: 1pt\">1.7.1. Ekvivalentas kopas<\/p>\n<p style=\"margin-left: 1pt\">A,B &#8211; kopas<\/p>\n<p style=\"margin-left: 1pt\">\n<p style=\"margin-left: 1pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija83.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 5pt\">1.7.2. Gal\u012bgas un bezgal\u012bgas kopas<\/p>\n<p style=\"margin-left: 5pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija84.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 5pt\">cardA, |A| &#8211; kopas elementu skaits<\/p>\n<p style=\"margin-left: 5pt\">|tuk\u0161a kopa| = 0<\/p>\n<p style=\"margin-left: 5pt\">A ~ {1,2,3,&#8230;,n} =&gt; |A|= n<\/p>\n<p style=\"margin-left: 5pt\">Kardin\u0101lskait\u013ci<\/p>\n<p style=\"margin-left: 5pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija85.png\" alt=\"\" \/><span style=\"font-family:Times New Roman; font-size:12pt\"><br \/>\n<\/span><\/p>\n<p style=\"margin-left: 5pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija86.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 5pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija87.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 5pt\">\n<p style=\"margin-left: 5pt\">Ja kopas A un B ir gal\u012bgas, tad apvienojums, \u0161\u0137\u0113lums, starp\u012bba utt. ir gal\u012bgas kopas<\/p>\n<p style=\"margin-left: 5pt\">Ja A un B ir bezgal\u012bgas, tad apvienojums ir bezgal\u012bgs, bet par p\u0101r\u0113j\u0101m op. neko nevar pateikt<\/p>\n<p style=\"margin-left: 5pt\">Ja A &#8211; gal\u012bga un B bezgal\u012bga, tad apvienojums ir bezgal\u012bgs, \u0161\u0137\u0113lums ir gal\u012bgs, starp\u012bba (B\\A) un simetrisk\u0101 starp\u012bba ir bezgal\u012bgas. Savuk\u0101rt A\\B &#8211; ir gal\u012bgs<\/p>\n<p style=\"margin-left: 5pt\">|AxB| = |A||B|<\/p>\n<p style=\"margin-left: 5pt\">2A- visas kopas A apak\u0161kopas.<\/p>\n<p style=\"margin-left: 5pt\">1.7.3. Sanumur\u0113jamas kopas<\/p>\n<p style=\"margin-left: 5pt\"><img src=\"https:\/\/ramuuns.com\/blog\/wp-content\/uploads\/2009\/10\/101209_1850_Koputeorija88.png\" alt=\"\" \/><\/p>\n<p style=\"margin-left: 5pt\">N, Z, Q &#8211; sanumur\u0113jamas<\/p>\n<p style=\"margin-left: 5pt\">R &#8211; nesanumur\u0113jama<\/p>\n<p style=\"margin-left: 5pt\">\n<p style=\"margin-left: 5pt\">A &#8211; gal\u012bga, B &#8211; sanumur\u0113jama. =&gt;<\/p>\n<div style=\"margin-left: 5pt\">\n<table style=\"border-collapse:collapse\" border=\"0\">\n<colgroup>\n<col style=\"width: 115px;\"><\/col>\n<col style=\"width: 111px;\"><\/col>\n<\/colgroup>\n<tbody>\n<tr>\n<td style=\"padding-top: 5px; padding-left: 5px; padding-bottom: 5px; padding-right: 5px; border-top:  solid #a3a3a3 1.0pt; border-left:  solid #a3a3a3 1.0pt; border-bottom:  solid #a3a3a3 1.0pt; border-right:  solid #a3a3a3 1.0pt\">gal\u012bgas<\/td>\n<td style=\"padding-top: 5px; padding-left: 5px; padding-bottom: 5px; padding-right: 5px; border-top:  solid #a3a3a3 1.0pt; border-left:  none; border-bottom:  solid #a3a3a3 1.0pt; border-right:  solid #a3a3a3 1.0pt\">A &amp; B, A\\B<\/td>\n<\/tr>\n<tr>\n<td style=\"padding-top: 5px; padding-left: 5px; padding-bottom: 5px; padding-right: 5px; border-top:  none; border-left:  solid #a3a3a3 1.0pt; border-bottom:  solid #a3a3a3 1.0pt; border-right:  solid #a3a3a3 1.0pt\">sanumur\u0113jamas<\/td>\n<td style=\"padding-top: 5px; padding-left: 5px; padding-bottom: 5px; padding-right: 5px; border-top:  none; border-left:  none; border-bottom:  solid #a3a3a3 1.0pt; border-right:  solid #a3a3a3 1.0pt\">AUB, B\\A, A^B<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p style=\"margin-left: 5pt\">A &#8211; sanumur\u0113jama, B &#8211; sanumur\u0113jama =&gt; AUB &#8211; sanumur\u0113jams<\/p>\n<p style=\"margin-left: 5pt\">\n","protected":false},"excerpt":{"rendered":"<p>Ieraksts rakstu s\u0113rij\u0101 lekciju pieraksti. 1.1 Kopas j\u0113dziens Def: kopa Ar v\u0101rdu kopa mat. saprot, t\u0101du j\u0113dzienu, kuram var viennoz\u012bm\u012bgi pateikt, ka k\u0101ds elements tai pieder vai nepieder Kopas parasti apz\u012bm\u0113 ar lielajiem lat\u012b\u0146u burtiem N &#8211; natur\u0101lie Z &#8211; veselie Q &#8211; racion\u0101lie R &#8211; re\u0101lie C &#8211; kompleksie \uf04c\uf020&#8211; tuk\u0161\u0101 kopa Interv\u0101li: (a,b) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[6],"tags":[],"_links":{"self":[{"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/posts\/72"}],"collection":[{"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/comments?post=72"}],"version-history":[{"count":2,"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/posts\/72\/revisions"}],"predecessor-version":[{"id":74,"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/posts\/72\/revisions\/74"}],"wp:attachment":[{"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/media?parent=72"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/categories?post=72"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ramuuns.com\/blog\/wp-json\/wp\/v2\/tags?post=72"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}