{"id":"a6e39199-1466-4895-8ba5-a342f6a02634","type":"https://serlo.org/editor","variant":"serlo-org","domainOrigin":"serlo.org","version":2,"editorVersion":"0.22.0","dateModified":"2025-02-26T18:24:44.975Z","document":{"plugin":"course","state":{"content":{"plugin":"rows","state":[]},"pages":[{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Einer der Grundbausteine, auf denen die modernen Kommunikations- und Informationstechnologien beruhen, ist Logik. Daher ist es höchst empfehlenswert, zumindest einen groben Blick auf die mathematische Logik zu werfen."}]}],"id":"dc4899b4-40f3-48c5-b3e9-9e8c056202ec"},{"plugin":"text","state":[{"type":"p","children":[{"text":"In diesem Kurs lernst du, dass eine Aussage nicht zwingend immer nur zum Deutschunterricht oder einer Gerichtsverhandlung gehört. Du untersuchst, ob Kombinationen oder Veränderungen von Aussagen gültig sind. Dieser Kurs bietet dabei nur einen kurzen und einfachen Einstieg in die einfachen Bausteine."}]}],"id":"387921e7-ca7b-4688-8a1c-d2ee26c8d8c3"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Insbesondere wirst du in diesem Kurs zunächst nur "},{"text":"Aussagen","em":true},{"text":" kennenlernen, das sind Behauptungen, deren Wahrheitsgehalt feststellbar ist. "}]},{"type":"p","children":[{"text":"In manchen Situationen ist das nicht ausreichend: willst du z.B. feststellen, ob der \"Schluss\" "},{"type":"math","src":"x<5\\Rightarrow x<6","inline":true,"children":[{"text":""}]},{"text":" gültig ist, musst du das Konzept der "},{"text":"Aussageformen","em":true},{"text":" verwenden - du kannst ja nicht feststellen, ob "},{"type":"math","src":"x<5","inline":true,"children":[{"text":""}]},{"text":" eine wahre Aussage ist, weil darin die Variable "},{"type":"math","src":"x","inline":true,"children":[{"text":""}]},{"text":" enthalten ist."}]}],"id":"5a62e10f-70c9-4e99-bd69-7e779da06c8a"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Eine kurze Einführung in Aussageformen findest du in den letzten Kapiteln des Kurses, eine ausführliche Diskussion dieses Themas in "},{"type":"a","href":"/44323","children":[{"text":"Grundlagen der Mathematik"}]},{"text":"."}]}],"id":"302eaf01-df79-44d5-9b9d-8c184f8ee707"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Hier lernst du, verschiedene Aussagen zu kombinieren und den Wahrheitsgehalt der Kombination abhängig von den Wahrheitsgehalten der einzelnen Teilaussagen zu bestimmen."}]}],"id":"e18d3f56-7da5-4f6a-9e9f-8a21562ef170"},{"plugin":"box","state":{"type":"note","title":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"dd3ec7be-92dc-417b-84d2-a340db75c2e5"},"anchorId":"box50089","content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"(Aussagen-)Logik untersucht nicht den Wahrheitsgehalt einer Aussage direkt, sondern leitet diesen aus bereits erlangtem Wissen ab."}]}],"id":"930f01c6-d455-478b-9f74-6237a303416e"}],"id":"4a217bae-4729-481d-b415-0c0beaff952e"}},"id":"fe151cb7-f2ce-4a48-a2c0-5034bfff88ab"}],"id":"64734b6c-a588-4d60-9391-5397fea3db09"},"id":"239508-course-page","title":"Einleitung (in Arbeit)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"In der Logik geht es darum, verschiedene Aussagen zu kombinieren."}]}],"id":"1c9fb449-0e94-4e65-ab33-9031f99e53a6"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Daher wird der zentrale Begriff der Aussage erklärt:"}]}],"id":"a832bec1-b3c6-4e3e-aff7-c72265579acf"},{"plugin":"box","state":{"type":"blank","title":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"2a699a75-2f2d-45fc-94d2-56df718491a1"},"anchorId":"box88167","content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Eine (mathematische) "},{"text":"Aussage","strong":true},{"text":" ist eine Behauptung über einen (mathematischen) Sachverhalt, von der feststellbar ist, ob sie wahr oder falsch ist."}]}],"id":"ff3a7cd0-8b47-416d-96a9-1ec6af0b238c"}],"id":"f41d0d25-9456-476c-a20d-40086ce71248"}},"id":"598c8ca4-3f7e-4dff-b58d-2ed140ad7663"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Das bedeutet, dass jeder Aussage genau einer der Wahrheitswerte wahr oder falsch zugeordnet werden kann. Dabei kommt es nicht darauf an, dass der Wahrheitswert direkt ersichtlich ist, sondern nur, dass dies prinzipiell möglich ist."}]}],"id":"13563be7-f5f7-43aa-a844-141e4de5afc7"},{"plugin":"box","state":{"type":"blank","title":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"912b8e81-8024-4c12-b6cb-4f411abc3710"},"anchorId":"box46272","content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Der Wahrheitswert \"wahr\" wird mit "},{"type":"math","src":"(w)","inline":true,"children":[{"text":"w"}]},{"text":", der Wert \"falsch\" mit "},{"type":"math","src":"(f)","inline":true,"children":[{"text":""}]},{"text":" abgekürzt."}]}],"id":"7841f057-4b79-4644-9816-6b7317c2eb9f"}],"id":"9535fc4a-6bd3-44ad-b9f7-fdd99967e7fe"}},"id":"cee7fcff-ef72-4c4d-858f-941264b430b8"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Beispiele","strong":true},{"text":" "}]}],"id":"2a2bddc9-c76b-47ac-b61d-ce6ad6de9f04"},{"plugin":"text","state":[{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"math","src":"1+1=1\\quad (w)","inline":true,"children":[{"text":""}]},{"text":""}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"math","src":"3+9<4\\quad (f)","inline":true,"children":[{"text":""}]},{"text":""}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"math","src":"f:x\\mapsto x^3","inline":true,"children":[{"text":""}]},{"text":" ist streng monoton steigend "},{"type":"math","src":"(w)","inline":true,"children":[{"text":""}]},{"text":""}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"math","src":"3^{100}>1000^{16}","inline":true,"children":[{"text":""}]},{"text":""}]}]}]}],"id":"d2b25930-ad4d-4630-a2f6-9bd140eab5eb"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Der vierten Aussage kann man nicht unbedingt direkt einen Wahrheitswert zuordnen. Es ist aber prinzipiell möglich: es ist "},{"type":"math","src":"3^{100}\\approx 5\\cdot 10^{47}","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"1000^{16}=10^{48}","inline":true,"children":[{"text":""}]},{"text":". Die Aussage ist also falsch."}]}],"id":"d0e318a4-a396-4888-a03b-9aa05ade6856"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Ein "},{"text":"Gegenbeispiel","strong":true},{"text":" ist \"Pudding schmeckt gut\", weil der Wahrheitsgehalt hiervon nicht objektiv bestimmbar ist."}]}],"id":"308c20b5-baa1-4980-a8d8-a693db2919c4"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Aussagen werden oft durch Großbuchstaben bezeichnet, also z.B."}]}],"id":"0c6f9245-6ce6-44ba-833e-870668045bc6"},{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"A:\\, 1+1=2","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"b2025259-ecb1-4fc5-bf9b-102f5eb63f34"}],"id":"93aca393-b520-4f2e-9b3f-199d0ae020de"},"id":"223934-course-page","title":"Aussagen und Wahrheitwert (in Arbeit)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Eine Aussage lässt sich einfach verneinen (man sagt auch \"negieren\"). Sprachlich geschieht das meistens durch das Wort \"nicht\":"}]}],"id":"fa270994-11a5-42b7-8dd3-eceaaacaab5f"},{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"A:","inline":true,"children":[{"text":"A:"}]},{"text":" \"Es regnet.\" wird nach der Verneinung zu "},{"type":"math","src":"\\neg A","inline":true,"children":[{"text":""}]},{"text":" (oder auch "},{"type":"math","src":"\\overline{A}","inline":true,"children":[{"text":""}]},{"text":"): \"Es regnet nicht.\" "}]},{"type":"p","children":[{"text":""},{"type":"math","src":"\\neg A","inline":true,"children":[{"text":""}]},{"text":" nennt man "},{"text":"Verneinung","strong":true},{"text":" oder "},{"text":"Negation","strong":true},{"text":" von "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":"."}]}],"id":"c07fb7e5-de95-4112-b0cf-16829b911525"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Das, was vorher gegolten hat, gilt jetzt genau nicht mehr. In Zweifelsfällen ist es oft am sichersten, der Aussage "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" den Satzteil \"Es ist nicht so, dass. . . \" voranzustellen."}]}],"id":"7cdcd46c-c68d-4d82-b711-265d4ee2f131"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Dabei gelten die Regeln:"}]}],"id":"8f3b95d6-ae75-48fd-8cd8-e321f265bc6a"},{"plugin":"text","state":[{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Ist "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" wahr, so ist "},{"type":"math","src":"\\neg A","inline":true,"children":[{"text":""}]},{"text":" falsch."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Ist "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" falsch, so ist "},{"type":"math","src":"\\neg A","inline":true,"children":[{"text":""}]},{"text":" wahr."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"math","src":"\\neg(\\neg A)","inline":true,"children":[{"text":""}]},{"text":" hat denselben Wahrheitswert wie "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":""}]}]}]}],"id":"5fd032c4-626a-4f44-9d3d-edc8b305f107"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die dritte Regel hat den Namen \""},{"text":"Gesetz der doppelten Negation","strong":true},{"text":"\"."}]}],"id":"843ad7a4-c3d2-4aa9-bd41-09a781f119c3"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Beispiele","strong":true}]}],"id":"36ec8924-ca2d-4b77-98a2-bab9e6c27d08"},{"plugin":"text","state":[{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Ist A die wahre Aussage \""},{"type":"math","src":"6","inline":true,"children":[{"text":"6"}]},{"text":" ist durch "},{"type":"math","src":"3","inline":true,"children":[{"text":"3"}]},{"text":" teilbar\", so ist "},{"type":"math","src":"\\neg A","inline":true,"children":[{"text":"¬A"}]},{"text":" die falsche Aussage \"Es ist nicht so, dass "},{"type":"math","src":"6 ","inline":true,"children":[{"text":"6"}]},{"text":" durch "},{"type":"math","src":"3","inline":true,"children":[{"text":"3"}]},{"text":" teilbar ist\". \nDas lässt sich auch als \""},{"type":"math","src":"6","inline":true,"children":[{"text":"6"}]},{"text":" ist nicht durch "},{"type":"math","src":"3","inline":true,"children":[{"text":"3"}]},{"text":" teilbar\" formulieren."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Die Negation von \""},{"type":"math","src":"5","inline":true,"children":[{"text":"5"}]},{"text":" ist eine gerade Zahl\" ist \""},{"type":"math","src":"5","inline":true,"children":[{"text":""}]},{"text":" ist eine ungerade Zahl\"."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Ist "},{"type":"math","src":"A","inline":true,"children":[{"text":"A"}]},{"text":" die (falsche) Aussage \"Alle Zahlen sind gerade\", so ist die Negation von "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" zunächst \"Es ist nicht so, dass alle Zahlen gerade sind\". Das kannst du zu \"Es gibt mindestens eine ungerade Zahl\" umformulieren.\nDie Aussage \"Alle Zahlen sind ungerade\" ist "},{"text":"nicht","strong":true},{"text":" die Negation von "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":", was man schon daran merken kann, dass diese Aussage genauso falsch wie "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" ist."}]}]}]}],"id":"e6716e72-42c6-47c2-8999-cca78a163697"}],"id":"e075b984-6475-4744-ac55-1f1b2745dcbd"},"id":"223936-course-page","title":"Nicht (in Arbeit)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Mehrere Aussagen lassen sich durch ein logisches \"Und\" (Symbol: "},{"type":"math","src":"\\wedge","inline":true,"children":[{"text":""}]},{"text":") miteinander verknüpfen. Das logische Und bedeutet dabei eigentlich \"und gleichzeitig\". Damit der Ausdruck als wahr ausgewertet wird, muss also "},{"text":"jede","strong":true},{"text":" der Aussagen wahr sein. "}]}],"id":"b2e1eafa-9021-43ff-8260-5892c21cac38"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Anders formuliert: Die Aussageverknüpfung "},{"type":"math","src":"A\\wedge B","inline":true,"children":[{"text":""}]},{"text":" ist schon falsch, wenn auch nur eine der beiden Aussagen "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":" falsch ist."}]}],"id":"9031f1d3-d5e3-42dd-9638-eaf5d065d02a"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die Aussageverknüpfung \"Und\" heißt auch "},{"text":"Konjunktion","strong":true},{"text":"."}]}],"id":"e0b27713-349e-4c6f-a759-7b789799f660"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Für "},{"type":"math","src":"A","inline":true,"children":[{"text":"A"}]},{"text":": \""},{"type":"math","src":"5<7","inline":true,"children":[{"text":""}]},{"text":"\" und "},{"type":"math","src":"B","inline":true,"children":[{"text":"B"}]},{"text":": \""},{"type":"math","src":"5","inline":true,"children":[{"text":""}]},{"text":" ist ungerade\" kann man "},{"type":"math","src":"A\\wedge B","inline":true,"children":[{"text":"AB"}]},{"text":" formulieren als \"5 ist kleiner als "},{"type":"math","src":"7","inline":true,"children":[{"text":"7"}]},{"text":" und ungerade\". Da "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":" wahr sind, ist auch "},{"type":"math","src":"A\\wedge B","inline":true,"children":[{"text":""}]},{"text":" wahr."}]}],"id":"38197590-7e8c-4f7d-91c9-97eea075bec9"},{"plugin":"text","state":[{"type":"p","children":[{"text":"In der Umgangssprache bezeichnet \"und\" oft eine zeitliche Abfolge:"}]},{"type":"p","children":[{"text":"\"Sie trank ihr Glas leer und ging\". "}]},{"type":"p","children":[{"text":"Mathematisch bedeutet "},{"type":"math","src":"A\\wedge B ","inline":true,"children":[{"text":"A∧B"}]},{"text":" das gleichzeitige Erfülltsein der beiden Aussagen."}]}],"id":"05b57854-9a2b-4864-a385-43da010dcbb2"},{"plugin":"box","state":{"type":"blank","title":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"8afd0d1e-b4eb-440e-b90f-117e820193e7"},"anchorId":"box36768","content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Ist "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" eine Aussage, dann ist "},{"type":"math","src":"A\\wedge \\neg A","inline":true,"children":[{"text":""}]},{"text":" immer falsch, weil entweder "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" oder "},{"type":"math","src":"\\neg A","inline":true,"children":[{"text":""}]},{"text":" falsch sein muss."}]}],"id":"291093b7-dce4-45c5-b903-50fbf95e282e"}],"id":"2c1727fa-9394-4a70-8ebe-b56d9eb71764"}},"id":"e4e93366-a173-4d00-99a8-5f0019099c8c"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Weitere Beispiele:","strong":true}]},{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"math","src":"\\mathbb{N}\\subset\\mathbb{Q}\\subset\\mathbb{R} ","inline":true,"children":[{"text":""}]},{"text":" ist die (wahre) Aussage "},{"type":"math","src":"(\\mathbb{N}\\subset\\mathbb{Q})\\wedge(\\mathbb{Q}\\subset\\mathbb{R})","inline":true,"children":[{"text":""}]},{"text":""}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Analog bedeutet "},{"type":"math","src":"3<\\pi<4","inline":true,"children":[{"text":""}]},{"text":" genauer "},{"type":"math","src":"(3<\\pi)\\wedge(\\pi<4)","inline":true,"children":[{"text":""}]},{"text":" - eine wahre Aussage, da beide Teile wahr sind."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":""},{"type":"math","src":"(1<2)\\wedge(4<3)","inline":true,"children":[{"text":""}]},{"text":" ist falsch, da der zweite Teil falsch ist."}]}]}]},{"type":"p","children":[{"text":""}]}],"id":"2f82e1a6-1d4a-4345-b46d-f3423570380a"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Rechenregeln","strong":true}]}],"id":"59cd81d5-b332-4988-b097-021929413c6e"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Es gilt das Assoziativ- und das Kommutativgesetz:"}]},{"type":"p","children":[{"text":""},{"type":"math","src":"A\\wedge B","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"B\\wedge A","inline":true,"children":[{"text":""}]},{"text":" haben denselben Wahrheitswert"}]},{"type":"p","children":[{"text":""},{"type":"math","src":"(A\\wedge B)\\wedge C","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"A\\wedge (B\\wedge C)","inline":true,"children":[{"text":""}]},{"text":" haben denselben Wahrheitswert. Diese Regel überträgt sich natürlich sofort auf mehr als drei Aussagen: Eine Und-Kombination von mehreren Aussagen ist nur dann wahr, wenn alle Aussagen wahr sind. Ist auch nur eine der Aussagen falsch, ist auch die Kombination falsch."}]}],"id":"c2165de9-3070-4883-bdfc-be41da75d25f"}],"id":"be4d6488-abfb-41e7-b355-c0aaffbbffa7"},"id":"223937-course-page","title":"Und (in Arbeit)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Das logische \"Oder\" "},{"type":"math","src":"\\vee","inline":true,"children":[{"text":""}]},{"text":" für zwei Aussagen kannst du dir vorstellen als: entweder die eine oder die andere oder beide Aussagen sind erfüllt. Es muss also "},{"text":"mindestens eine","strong":true},{"text":" der beteiligten Aussagen gelten, damit der Ausdruck als wahr gewertet werden kann. "}]}],"id":"f3c52228-4306-4b81-ba40-7823dfd516b2"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Anders formuliert: die Verknüpfung "},{"type":"math","src":"A\\vee B","inline":true,"children":[{"text":""}]},{"text":" ist nur dann falsch, wenn "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":" beide falsch sind."}]}],"id":"f8f62532-ebf5-48b5-9b22-fbb6d0e88631"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die Aussageverknüpfung \"Oder\" heißt auch "},{"text":"Disjunktion","strong":true},{"text":"."}]}],"id":"5abe703c-0f67-4853-a0db-d7ffbeac0bce"},{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"(2<3)\\vee (4<3)","inline":true,"children":[{"text":""}]},{"text":" ist wahr, da der erste Teil wahr ist. Dass der zweite Teil falsch ist, ändert nichts."}]}],"id":"00e6bf7e-a743-46de-a0dc-b31ece1854e5"},{"plugin":"box","state":{"type":"blank","title":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"10a1127f-afb3-48d2-be9a-48c4a2654fe9"},"anchorId":"box78501","content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Ist "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" eine Aussage, dann ist "},{"type":"math","src":"A\\vee \\neg A","inline":true,"children":[{"text":""}]},{"text":" immer wahr, weil entweder "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" oder "},{"type":"math","src":"\\neg A","inline":true,"children":[{"text":""}]},{"text":" wahr sein muss."}]}],"id":"ddcf7f57-7eb4-4bed-91c6-927377c5d9b7"}],"id":"70602f34-1516-4e25-8363-7a774ad38f02"}},"id":"42a68f2c-8645-4e4b-a82b-b42e4be6e5eb"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die (nicht ganz so mathematische) Aussage \"Sein oder nicht Sein\" ist also unproblematisch, weil sie immer wahr ist."}]}],"id":"7fd08887-6ae6-4e04-b869-cc25809c02b2"},{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"(3<2)\\vee (2^2>5)","inline":true,"children":[{"text":""}]},{"text":" ist falsch, weil beide Klammern falsch sind."}]}],"id":"42eb9719-a99c-4c0c-b452-a67ec0dfd7d9"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Rechenregeln","strong":true}]}],"id":"e8df5d86-4675-401e-ab5d-59d27e66ff8e"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Es gilt das Assoziativ- und das Kommutativgesetz:"}]},{"type":"p","children":[{"text":""},{"type":"math","src":"A\\vee B","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"B\\vee A","inline":true,"children":[{"text":""}]},{"text":" haben denselben Wahrheitswert"}]},{"type":"p","children":[{"text":""},{"type":"math","src":"(A\\vee B)\\vee C","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"A\\vee (B\\vee C)","inline":true,"children":[{"text":""}]},{"text":" haben denselben Wahrheitswert. Diese Regel überträgt sich natürlich sofort auf mehr als drei Aussagen: eine Oder-Verknüpfung ist wahr, wenn mindestens eine der Aussagen wahr ist. Anders formuliert: eine Oder-Verknüpfung ist nur dann falsch, wenn alle Aussagen falsch sind."}]}],"id":"49228019-1544-4989-bb69-5df7b8dc98c6"}],"id":"66538e2b-6e1f-45fd-b82e-f5ad3254b95e"},"id":"223938-course-page","title":"Oder (in Arbeit)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Um dir einen Überblick über die Bedeutung von Aussagekombinationen zu verschaffen, sind Wahrheitswerttabellen ein gutes Mittel."}]}],"id":"fa107992-2e89-4a38-ac0e-8f07c6f2290b"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Dazu schreibst du (möglichst systematisch) alle möglichen Wahrheitswerte der beteiligten Aussagen in eine Spalte und untersuchst den Wahrheitswert einer Kombination."}]},{"type":"p","children":[{"text":"Je nach Anzahl der Aussagen musst du eine passende Anzahl von Fällen unterscheiden. 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Aber auch rechts kann man nach dem Absorptionsgesetz "},{"type":"math","src":"A\\vee B","inline":true,"children":[{"text":""}]},{"text":" zu "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"A\\vee C","inline":true,"children":[{"text":""}]},{"text":" zu "},{"type":"math","src":"C","inline":true,"children":[{"text":""}]},{"text":" zusammenziehen, also ist die rechte Seite auch zu "},{"type":"math","src":"B\\vee C","inline":true,"children":[{"text":""}]},{"text":" äquivalent."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Wenn "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" falsch ist, dann ist sicher die linke Seite falsch, und weil dann auch "},{"type":"math","src":"A\\wedge B","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"A\\wedge C","inline":true,"children":[{"text":""}]},{"text":" falsch sind, ist es auch die rechte Seite."}]}]}]},{"type":"p","children":[{"text":""}]}],"id":"a837acaf-48c6-4e00-8d2f-b1e66c1a8a11"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Ganz ähnlich beweist du das zweite Distributivgesetz"}]}],"id":"c5497953-edd0-4c5b-91b1-d8cc5001374d"},{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"A\\vee(B\\wedge C)\\Leftrightarrow (A\\vee B)\\wedge (A\\vee C)","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"4ea71c65-d887-44c7-8f8f-9568e5025a79"},{"plugin":"text","state":[{"type":"unordered-list","children":[{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Wenn "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" wahr ist, dann ist die linke Seite wahr, und weil dann auch "},{"type":"math","src":"A\\vee B","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"A\\vee C","inline":true,"children":[{"text":""}]},{"text":" wahr sind, ist es auch die rechte Seite."}]}]},{"type":"list-item","children":[{"type":"list-item-child","children":[{"text":"Wenn "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" falsch ist, ist nach dem Absorptionsgesetz die linke Seite äquivalent zu "},{"type":"math","src":"B\\wedge C","inline":true,"children":[{"text":""}]},{"text":". Aus dem gleichen Grund ist "},{"type":"math","src":"A\\vee B","inline":true,"children":[{"text":""}]},{"text":" zu "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"A\\vee C","inline":true,"children":[{"text":""}]},{"text":" zu "},{"type":"math","src":"C","inline":true,"children":[{"text":""}]},{"text":" äquivalent, also ist die rechte Seite genau wie die linke zu "},{"type":"math","src":"B\\wedge C","inline":true,"children":[{"text":""}]},{"text":" äquivalent."}]}]}]}],"id":"1777017e-9d31-4200-846c-d43ea541da64"}],"id":"920e625b-4efb-4dbc-8f66-894423e0cc6a"},"id":"239795-course-page","title":"Distributivgesetze (in Arbeit)"},{"content":{"plugin":"rows","state":[{"plugin":"text","state":[{"type":"p","children":[{"text":"Sind "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" und "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":" Aussagen, so definiert man die "},{"text":"Implikation","strong":true},{"text":" "},{"type":"math","src":"A\\Rightarrow B","inline":true,"children":[{"text":""}]},{"text":" durch "},{"type":"math","src":"\\neg A \\vee B","inline":true,"children":[{"text":""}]},{"text":"."}]}],"id":"0d01c9f3-cb4a-4c0a-ba11-df252a068675"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die Idee dabei ist so: die Implikation soll bedeuten, dass aus der Aussage "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" die Aussage "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":" folgt, d.h. wenn "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" wahr ist, soll auch "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":" wahr sein. Ist "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" falsch, gibt es keine Bedingung an "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":"."}]}],"id":"23e31483-1327-4eee-91c8-7ee35b7644dd"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Das bedeutet, dass die Wahrheitswerttabelle von "},{"type":"math","src":"A\\Rightarrow B","inline":true,"children":[{"text":""}]},{"text":" so aussieht:"}]}],"id":"f08b89d2-2b1a-426f-bb71-8d5de7e413fe"},{"plugin":"serloTable","state":{"rows":[{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"ee35722b-aee4-439e-a989-37b31c2e2037"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"d2a94136-0018-46e0-af09-ab3f5ea7dc58"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"8e0d86d2-79e8-4b94-8d3f-e0f06eaa6303"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"A\\Rightarrow B","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"a2541e63-763f-461f-8040-d095c78b2a58"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"w","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"8d8064e5-2a0e-45a1-8afa-d26eb95fe7e9"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"w","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"0a22fbca-3b29-4bcb-b10a-f532b38feb74"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"53492641-20fa-4ba8-bb7d-aeebe4630c3c"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"w","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"fa5c6944-537c-49bc-af0e-032a9a9a2228"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"w","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"b5c107df-0e40-46c1-95c1-a43cd0589a2d"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"f","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"435dc6a2-84e9-4f51-a375-f3f9baa25b93"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"c309007c-4269-45de-8ba7-a3775f84bb7f"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"f","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"4a7fb145-fb26-4ac0-ba93-0187e3d56f57"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"f","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"d0e4031b-b35d-499b-95cc-788a79453fd6"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"w","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"9fddb606-5bac-4ecb-9e0c-622b82c04586"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"03bcd51d-3a27-4fec-83eb-d4d04a099636"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"w","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"7501845a-d2c7-4604-894b-d7c0a38d7079"}}]},{"columns":[{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"f","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"96d2b2bf-2311-4567-a22e-5b222aa2cffa"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"f","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"aecede9e-e268-43c3-a679-259d5c622c4d"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{}]}],"id":"e973b136-eb22-40aa-9bdb-728b4e69cd2a"}},{"content":{"plugin":"text","state":[{"type":"p","children":[{"text":""},{"type":"math","src":"w","inline":true,"children":[{"text":""}]},{"text":""}]}],"id":"da49d59e-8c92-4d7b-ad0a-29920a5f3c0e"}}]}],"tableType":"OnlyColumnHeader"},"id":"3997e07e-62d2-4c26-b864-3853177a86d8"},{"plugin":"text","state":[{"type":"p","children":[{"text":"Die ersten beiden Zeilen sagen \"Wenn "},{"type":"math","src":"A","inline":true,"children":[{"text":""}]},{"text":" wahr ist, ist auch "},{"type":"math","src":"B","inline":true,"children":[{"text":""}]},{"text":" wahr und nicht etwa falsch\". 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