Disjunkte Vereinigung Beispiel Essay

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Disjunkte Mengen[Bearbeiten]

Erklärung und Beispiele[Bearbeiten]

Zwei Mengen und , die keine gemeinsamen Elemente besitzen, nennt man . Für disjunkte Mengen gibt es auch die Bezeichnungen elementfremd oder durchschnittsfremd. Das Wort „disjunkt“ leitet sich dabei vom lateinischen Wort „disiunctum“ ab, was soviel wie „getrennt“ bedeutet. Nehme als Beispiel die folgenden zwei Mengen

Diese beiden Mengen sind nicht disjunkt, weil sie das „Klavier“ als gemeinsames Objekt besitzen. Demgegenüber sind aber die folgenden beiden Mengen disjunkt, weil sie keine gemeinsamen Elemente besitzen:

Weitere Beispiele:

  • Die Menge mit der Spielkarte und dem Buch ist disjunkt zur Menge mit der Gitarre und der Trommel.

  • Ein weiteres Beispiel für zwei disjunkte Mengen

  • Disjunkte Mengen im Mengendiagramm

Beschreibung über Schnittmenge[Bearbeiten]

Später werden wir sehen, dass die Menge aller gemeinsamen Elemente von und ist. Zwei Mengen sind also genau dann disjunkt, wenn die Menge kein Element besitzt. Eine solche Menge ohne Elemente nennt man „leere Menge“, welche als notiert wird. Es ist also:

Definition und Verständnisfragen[Bearbeiten]

Definition 

Zwei Mengen nennt man disjunkt, wenn sie keine gemeinsamen Elemente besitzen:

Verständnisfrage: Welche der folgenden Paare von Mengen sind disjunkt?

  1. und
  2. und
  3. und
  4. und
  5. und

Antwort:

  1. und sind nicht disjunkt, weil sie beide als Element besitzen.
  2. und sind disjunkt, weil sie keine gemeinsamen Elemente besitzen.
  3. und sind nicht disjunkt, weil sie zum Beispiel die Zahl als gemeinsames Element besitzen.
  4. und sind disjunkt, weil ist.
  5. und sind nicht disjunkt, weil sie als gemeinsames Element besitzen.

Folgende Fragen setzen voraus, dass du den Begriff der Schnittmenge und der leeren Menge schon kennst (siehe die nächsten Kapitel). Du kannst diese Fragen also gerne überspringen, wenn du diese Begriffe noch nicht kennen solltest.

Verständnisfrage: Mit welchen Mengen ist disjunkt?

ist mit jeder Menge disjunkt, weil ist.

Verständnisfrage: Wann ist eine Menge zu sich selbst disjunkt?

Eine Menge ist genau dann zu sich selbst disjunkt, wenn ist. Wegen ist dies genau dann der Fall, wenn ist:

Verständnisfrage: Nehme die zwei Mengen und . Unter welchen Umständen sind diese beiden Mengen disjunkt?

Diese beiden Mengen sind genau dann disjunkt, wenn ist.

Paarweise disjunkte Mengensysteme[Bearbeiten]

Ein Mengensystem , also eine Menge von Mengen, nennt man , wenn jeweils zwei verschiedene Mengen disjunkt sind. Egal welche zwei unterschiedlichen Mengen und aus ausgewählt werden, diese beiden Mengen besitzen keine gemeinsamen Elemente. Zum Beispiel ist folgendes Mengensystem paarweise disjunkt:

Demgegenüber ist das folgende Mengensystem nicht paarweise disjunkt, weil sich die Mengen und überschneiden:

Wir fassen zusammen:

Definition 

Ein Mengensystem ist paarweise disjunkt, wenn alle verschiedenen Mengen disjunkt sind. Es gilt also:

Disjunkte Mengen – Erklärung und Definition (Video vom Podcast The Wicked Mu)
Paarweise disjunkte Mengensysteme (Video vom Podcast The Wicked Mu)
Beispiel eines paarweise disjunkten Mengensystems mit drei Mengen

Verständnisfrage: Sei ein Mengensystem besteht aus nur einer Menge

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