{"id":556,"date":"2017-05-29T23:11:14","date_gmt":"2017-05-29T21:11:14","guid":{"rendered":"http:\/\/theory-of-science.com\/de\/?page_id=556"},"modified":"2021-02-04T16:23:47","modified_gmt":"2021-02-04T15:23:47","slug":"uebung-12-05","status":"publish","type":"page","link":"https:\/\/theory-of-science.com\/de\/uebungen\/abschnitt-12\/uebung-12-05\/","title":{"rendered":"\u00dc12-5: Bijektion von kartesischen Produkten"},"content":{"rendered":"\n<p>Eine Bijektion von einem kartesischen Produkt <em>X<\/em>&nbsp;\u00d7 <em>Y<\/em> zu <em>X<\/em> &#8218;&nbsp;\u00d7 <em>Y<\/em> &#8218; wird normalerweise so definiert, dass zun\u00e4chst Bijektionen <em>f <\/em>: <em>X<\/em>&nbsp;\u2192 <em>X<\/em> &#8218; und <em>f<\/em> &#8218; : <em>Y<\/em>&nbsp;\u2192 <em>Y<\/em> &#8218; definiert werden.<\/p>\n\n\n\n<p><strong>a)<\/strong> <em>f<\/em> : <em>X<\/em>&nbsp;\u2192 <em>X<\/em> &#8218; und <em>f<\/em> &#8218; : <em>Y<\/em>&nbsp;\u2192 <em>Y<\/em> &#8218; seien gegeben. Definieren Sie eine Funktion <em>g<\/em> : ( <em>X<\/em>&nbsp;\u00d7 <em>X<\/em> &#8218; )&nbsp;\u2192 ( <em>Y<\/em>&nbsp;\u00d7 <em>Y<\/em> &#8218; ) und beweisen Sie, dass <em>g<\/em> bijektiv ist. (Hinweis: <em>g<\/em> ( <em>x<\/em>, <em>y<\/em> ) =&nbsp;\u2329 <em>f<\/em> ( <em>x<\/em> ), <em>f<\/em> &#8218; ( <em>y<\/em> ) \u232a )<\/p>\n\n\n\n<p><strong>b)<\/strong> Sei <em>A<\/em> die Menge { 1, 2, 3 } und <em>B<\/em> die Menge { <em>Uta<\/em>, <em>Udo<\/em>, <em>Peter<\/em> }. Bilden Sie die kartesischen Produkte <em>A<\/em>&nbsp;\u00d7 <em>A<\/em> und <em>B<\/em>&nbsp;\u00d7 <em>B<\/em>. Definieren Sie eine bijektive Funktion von&nbsp;<em>A<\/em>&nbsp;\u00d7 <em>A<\/em> nach <em>B<\/em>&nbsp;\u00d7 <em>B<\/em>.<br><\/p>\n\n\n<ul class=\"nav nav-pills nav-justified\">\n<li><a href=\"https:\/\/theory-of-science.com\/de\/uebungen\/abschnitt-12\/uebung-12-04\/\">&lt;&lt;&lt;<\/a><\/li>\n<li><a href=\"https:\/\/theory-of-science.com\/de\/uebungen\/abschnitt-12\/uebung-12-06\/\">&gt;&gt;&gt;<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Eine Bijektion von einem kartesischen Produkt X&nbsp;\u00d7 Y zu X &#8218;&nbsp;\u00d7 Y &#8218; wird normalerweise so definiert, dass zun\u00e4chst Bijektionen f : X&nbsp;\u2192 X &#8218; und f &#8218; : Y&nbsp;\u2192 Y &#8218; definiert werden. a) f : X&nbsp;\u2192 X &#8218; und f &#8218; : Y&nbsp;\u2192 Y &#8218; seien gegeben. Definieren Sie eine Funktion g : [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":67,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"page-fullwidth.php","meta":{"footnotes":""},"class_list":["post-556","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/theory-of-science.com\/de\/wp-json\/wp\/v2\/pages\/556","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/theory-of-science.com\/de\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/theory-of-science.com\/de\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/theory-of-science.com\/de\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theory-of-science.com\/de\/wp-json\/wp\/v2\/comments?post=556"}],"version-history":[{"count":8,"href":"https:\/\/theory-of-science.com\/de\/wp-json\/wp\/v2\/pages\/556\/revisions"}],"predecessor-version":[{"id":4461,"href":"https:\/\/theory-of-science.com\/de\/wp-json\/wp\/v2\/pages\/556\/revisions\/4461"}],"up":[{"embeddable":true,"href":"https:\/\/theory-of-science.com\/de\/wp-json\/wp\/v2\/pages\/67"}],"wp:attachment":[{"href":"https:\/\/theory-of-science.com\/de\/wp-json\/wp\/v2\/media?parent=556"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}