Aufgaben:Exercise 1.2Z: Sets of Digits - Revision history

Guenter at 13:13, 25 November 2021

2021-11-25T13:13:01Z

Guenter at 12:52, 25 November 2021

2021-11-25T12:52:07Z

Noah at 15:26, 23 November 2021

2021-11-23T15:26:06Z

Noah at 15:23, 23 November 2021

2021-11-23T15:23:46Z

Noah at 15:11, 23 November 2021

2021-11-23T15:11:35Z

Noah at 15:11, 23 November 2021

2021-11-23T15:11:08Z

Noah at 21:28, 21 November 2021

2021-11-21T21:28:11Z

Noah at 21:58, 20 November 2021

2021-11-20T21:58:03Z

Noah at 18:41, 20 November 2021

2021-11-20T18:41:43Z

Guenter at 15:47, 12 October 2021

2021-10-12T15:47:04Z

@@ Line 61: / Line 61: @@
 :$$ E = (A \cup B) \cap (\overline A \cup \overline B) = (A \cap \overline A) \cup (A \cap \overline B) \cup (\overline A \cap B) \cup (\overline A \cap \overline B) = (A \cap \overline B) \cup (\overline A \cap B) = D = \{1, 2, 6, 9\},$$
-:$$F = (A \cup C= \cap \overline B = \{1, 2, 3, 5, 6, 7, 8\} \cap \{1, 2, 4, 5, 7, 8\} = \{1, 2, 5, 7, 8\},$$
+:$$F = (A \cup C) \cap \overline B = \{1, 2, 3, 5, 6, 7, 8\} \cap \{1, 2, 4, 5, 7, 8\} = \{1, 2, 5, 7, 8\},$$
 :$$H = (\bar A \cap \overline C) \cup (A \cap B \cap C) = (\overline A \cap \overline C) \cup \phi = \{4, 9\}.$$
-'''(1)'''&nbsp; Only the <u>proposed solution 2</u> is correct:
+'''(1)'''&nbsp; Only the&nbsp; <u>proposed solution 2</u>&nbsp; is correct:
 * $A$&nbsp; and&nbsp; $C$&nbsp; have no common element.
 * $A$&nbsp; and&nbsp; $B$&nbsp; each contain a&nbsp; $3$.
@@ Line 72: / Line 72: @@
-'''(2)'''&nbsp; Correct is the <u>proposed solution 2</u>:
+'''(2)'''&nbsp; Correct is the&nbsp; <u>proposed solution 2</u>:
 *No digit is contained in&nbsp; $A$,&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; at the same time &nbsp; &rArr; &nbsp;  $ A \cap B \cap C = \phi$ &nbsp; &rArr; &nbsp; $ \overline{A \cap B \cap C} = \overline{\phi} = G$.
-*The first proposition, on the other hand, is wrong. It is missing a&nbsp; $4$.
+*The first proposition, on the other hand,&nbsp; is wrong.&nbsp; It is missing a&nbsp; $4$.
-'''(3)'''&nbsp; Correct are the <u>proposed solution 1,  2 and 4</u>:
+'''(3)'''&nbsp; Correct are the&nbsp; <u>proposed solutions 1,  2 and 4</u>:
 *The first proposal is correct: &nbsp; The sets&nbsp; $D$&nbsp; and&nbsp; $E$&nbsp; contain exactly the same elements and thus also their complementary sets.
-*The second proposal is also correct: &nbsp; In general, i.e. for any&nbsp; $X$&nbsp; and&nbsp; $B$&nbsp; the following holds:&nbsp; $X \cap \overline B \subset \overline B \ \Rightarrow$ &nbsp; With $X = A \cup C$ it follows that $F \subset \overline B$.
+*The second proposal is also correct: &nbsp; In general, i.e. for any&nbsp; $X$&nbsp; and&nbsp; $B$&nbsp; the following holds:&nbsp; $(X \cap \overline B) \subset \overline B \ \Rightarrow$ &nbsp; With $X = A \cup C$ it follows that $F \subset \overline B$.
-*The last proposal is also correct: &nbsp; $A = \{1, 2, 3\},$&nbsp;  $C = \{5, 6, 7, 8\}$&nbsp; and&nbsp; $H = \{4, 9\}$ form a "complete system".
+*The last proposal is also correct: &nbsp; $A = \{1, 2, 3\},$&nbsp;  $C = \{5, 6, 7, 8\}$&nbsp; and&nbsp; $H = \{4, 9\}$&nbsp; form a "complete system".
-*The third suggestion, on the other hand, is wrong because&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; are not disjoint.
+*The third suggestion,&nbsp; on the other hand,&nbsp; is wrong because&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; are not disjoint.
 {{ML-Fuß}}

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Theory_of_Stochastic_Signals/Set_Theory_Basics}}
-[[File:EN_Sto_Z_1_2_neu.png|right|frame|Sets of digits&nbsp; $A$,&nbsp; $B$,&nbsp; $C$]]
+[[File:EN_Sto_Z_1_2_neu.png|right|frame|Sets of digits:&nbsp; $A$,&nbsp; $B$,&nbsp; $C$]]
 Let the universal set&nbsp; $G$&nbsp; be the set of all digits between&nbsp; $1$&nbsp; and&nbsp; $9$.&nbsp; Given are the following subsets:
 :$$A = \big[\text{digits} \leqslant 3\big],$$
 :$$ B = \big[\text{digits divisible by 3}\big],$$
-:$$ C = \big[\text{digits 5, 6, 7, 8}\big],$$
+:$$ C = \big[\text{digits 5, 6, 7, 8}\big].$$
-Besides these, let other sets be defined:
+Besides these,&nbsp; let other sets be defined:
 :$$D = (A \cap \overline B) \cup (\overline A \cap B),$$
 :$$E = (A \cup B) \cap (\overline A \cup \overline B), $$
 :$$F = (A \cup C) \cap \overline B, $$
-:$$G = (\overline A \cap \overline C) \cup (A \cap B \cap C).$$
+:$$H = (\overline A \cap \overline C) \cup (A \cap B \cap C).$$
 First consider which digits belong to the sets&nbsp; $D$,&nbsp; $E$,&nbsp; $F$&nbsp; and&nbsp; $H$&nbsp;  and then answer the following questions. <br>Justify your answers in terms of set theory.
@@ Line 25: / Line 21: @@
 Hints:
-*The task belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics|Set theory basics]].
+*The task belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics|Set Theory Basics]].
-*The topic of this chapter is illustrated with examples in the  (German language)  learning video [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)|Mengentheoretische Begriffe und Gesetzmäßigkeiten]] $\Rightarrow$ Set Theoretical Concepts and Laws.
+*The topic of this chapter is illustrated with examples in the&nbsp;  (German language)&nbsp;  learning video
@@ Line 44: / Line 42: @@
 + The complementary set to&nbsp; $A \cap B \cap C$&nbsp; gives the universal set&nbsp; $G$.
-{Which of the following statements is correct?
+{Which of the following statements are correct?
 |type="[]"}
 + The complementary sets of&nbsp; $D$&nbsp; and&nbsp; $E$&nbsp; are identical.
-+ $F$&nbsp; is a subset of the complementary set of &nbsp; $B$.
++ $F$&nbsp; is a subset of the complementary set of&nbsp; $B$.
 - The sets&nbsp; $B$,&nbsp; $C$&nbsp; and&nbsp; $D$&nbsp; form a complete system.
 + The sets&nbsp; $A$,&nbsp; $C$&nbsp; and&nbsp; $H$&nbsp; form a complete system.

← Older revision		Revision as of 15:26, 23 November 2021
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	*The task belongs to the chapter  [[Theory_of_Stochastic_Signals/Set_Theory_Basics\|Set theory basics]].		*The task belongs to the chapter  [[Theory_of_Stochastic_Signals/Set_Theory_Basics\|Set theory basics]].

−	*The topic of this chapter is illustrated with examples in the (German language) learning video [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]] $\Rightarrow$.	+	*The topic of this chapter is illustrated with examples in the (German language) learning video [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]] $\Rightarrow$ Set Theoretical Concepts and Laws.

← Older revision		Revision as of 15:23, 23 November 2021
Line 27:		Line 27:
	*The task belongs to the chapter  [[Theory_of_Stochastic_Signals/Set_Theory_Basics\|Set theory basics]].		*The task belongs to the chapter  [[Theory_of_Stochastic_Signals/Set_Theory_Basics\|Set theory basics]].

−	*The topic of this chapter is illustrated with examples in the (German language) learning video[[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]] \Rightarrow.	+	*The topic of this chapter is illustrated with examples in the (German language) learning video [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)\|Mengentheoretische Begriffe und Gesetzmäßigkeiten]] $\Rightarrow$.

← Older revision		Revision as of 15:11, 23 November 2021
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−	{{quiz-Header\|Buchseite=~~Stochastische Signaltheorie~~/~~Mengentheoretische Grundlagen~~}}	+	{{quiz-Header\|Buchseite=Theory_of_Stochastic_Signals/Set_Theory_Basics}}

	[[File:EN_Sto_Z_1_2_neu.png\|right\|frame\|Sets of digits  $A$,  $B$,  $C$]]		[[File:EN_Sto_Z_1_2_neu.png\|right\|frame\|Sets of digits  $A$,  $B$,  $C$]]

@@ Line 30: / Line 30: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Welche der folgenden Aussagen sind richtig?
+{Which of the following statements are correct?
 |type="[]"}
-- $A$&nbsp; und&nbsp; $B$&nbsp; sind disjunkte Mengen.
+- $A$&nbsp; and&nbsp; $B$&nbsp; are disjoint sets.
-+ $A$&nbsp; und&nbsp; $C$&nbsp; sind disjunkte Mengen.
++ $A$&nbsp; and&nbsp; $C$&nbsp; are disjoint sets.
-- $B$&nbsp; und&nbsp; $C$&nbsp; sind disjunkte Mengen.
+- $B$&nbsp; and&nbsp; $C$&nbsp; are disjoint sets.
-{Welche der folgenden Aussagen sind richtig?
+{Which of the following statements are correct?
 |type="[]"}
-- Die Vereinigungsmenge&nbsp; $A \cup B \cup C$&nbsp; ergibt die Grundmenge&nbsp; $G$.
+- The union&nbsp; $A \cup B \cup C$&nbsp; gives the universal set&nbsp; $G$.
-+ Die Komplementärmenge zu&nbsp; $A \cap B \cap C$&nbsp; ergibt die Grundmenge&nbsp; $G$.
++ The complementary set to&nbsp; $A \cap B \cap C$&nbsp; gives the universal set&nbsp; $G$.
-{Welche der folgenden Aussagen sind richtig?
+{Which of the following statements is correct?
 |type="[]"}
-+ Die Komplementärmengen von&nbsp; $D$&nbsp; und&nbsp; $E$&nbsp; sind identisch.
++ The complementary sets of&nbsp; $D$&nbsp; and&nbsp; $E$&nbsp; are identical.
-+ $F$&nbsp; ist eine Teilmenge der Komplementärmenge von&nbsp; $B$.
++ $F$&nbsp; is a subset of the complementary set of &nbsp; $B$.
-- Die Mengen&nbsp; $B$,&nbsp; $C$&nbsp; und&nbsp; $D$&nbsp; bilden ein vollständiges System.
+- The sets&nbsp; $B$,&nbsp; $C$&nbsp; and&nbsp; $D$&nbsp; form a complete system.
-+ Die Mengen&nbsp; $A$,&nbsp; $C$&nbsp; und&nbsp; $H$&nbsp; bilden ein vollständiges System.
++ The sets&nbsp; $A$,&nbsp; $C$&nbsp; and&nbsp; $H$&nbsp; form a complete system.
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-Für die weiteren in der Aufgabe definierten Mengen gilt:
+For the other sets defined in the problem holds:
 :$$ D = (A \cap \overline B) \cup (\overline A \cap B)
@@ Line 67: / Line 67: @@
 :$$H = (\bar A \cap \overline C) \cup (A \cap B \cap C) = (\overline A \cap \overline C) \cup \phi = \{4, 9\}.$$
-'''(1)'''&nbsp; Richtig ist nur der <u>Lösungsvorschlag 2</u>:
+'''(1)'''&nbsp; Only the <u>proposed solution 2</u> is correct:
-* $A$&nbsp; und&nbsp; $C$&nbsp; haben kein gemeinsames Element.
+* $A$&nbsp; and&nbsp; $C$&nbsp; have no common element.
-* $A$&nbsp; und&nbsp; $B$&nbsp; beinhalten jeweils die&nbsp; $3$.
+* $A$&nbsp; and&nbsp; $B$&nbsp; each contain a&nbsp; $3$.
-*$B$&nbsp; und&nbsp; $C$&nbsp; beinhalten jeweils die&nbsp; $6$.
+* $B$&nbsp; and&nbsp; $C$&nbsp; each contain a&nbsp; $6$.
-'''(2)'''&nbsp; Richtig ist der <u>Lösungsvorschlag 2</u>:
+'''(2)'''&nbsp; Correct is the <u>proposed solution 2</u>:
-*Keine Ziffer ist gleichzeitig in&nbsp; $A$,&nbsp; $B$&nbsp; und&nbsp; $C$&nbsp; enthalten &nbsp; &rArr; &nbsp;  $ A \cap B \cap C = \phi$ &nbsp; &rArr; &nbsp; $ \overline{A \cap B \cap C} = \overline{\phi} = G$.
+*No digit is contained in&nbsp; $A$,&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; at the same time &nbsp; &rArr; &nbsp;  $ A \cap B \cap C = \phi$ &nbsp; &rArr; &nbsp; $ \overline{A \cap B \cap C} = \overline{\phi} = G$.
-*Der erste Vorschlag ist dagegen falsch. Es fehlt die&nbsp; $4$.
+*The first proposition, on the other hand, is wrong. It is missing a&nbsp; $4$.
-'''(3)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 1,  2 und 4</u>:
+'''(3)'''&nbsp; Correct are the <u>proposed solution 1,  2 and 4</u>:
-*Der erste Vorschlag ist richtig: &nbsp; Die Mengen&nbsp; $D$&nbsp; und&nbsp; $E$&nbsp; enthalten genau die gleichen Elemente und somit auch deren Komplementärmengen.
+*The first proposal is correct: &nbsp; The sets&nbsp; $D$&nbsp; and&nbsp; $E$&nbsp; contain exactly the same elements and thus also their complementary sets.
-*Auch der zweite Vorschlag ist richtig: &nbsp; Allgemein, das heißt für beliebige&nbsp; $X$&nbsp; und&nbsp; $B$&nbsp; gilt:&nbsp; $X \cap \overline B \subset \overline B \ \Rightarrow$ &nbsp; Mit $X = A \cup C$ folgt somit $F \subset \overline B$.
+*The second proposal is also correct: &nbsp; In general, i.e. for any&nbsp; $X$&nbsp; and&nbsp; $B$&nbsp; the following holds:&nbsp; $X \cap \overline B \subset \overline B \ \Rightarrow$ &nbsp; With $X = A \cup C$ it follows that $F \subset \overline B$.
-*Auch der letzte Vorschlag ist richtig: &nbsp; $A = \{1, 2, 3\},$&nbsp;  $C = \{5, 6, 7, 8\}$&nbsp; und&nbsp; $H = \{4, 9\}$ bilden ein "vollständiges System".
+*The last proposal is also correct: &nbsp; $A = \{1, 2, 3\},$&nbsp;  $C = \{5, 6, 7, 8\}$&nbsp; and&nbsp; $H = \{4, 9\}$ form a "complete system".
-*Der dritte Vorschlag ist dagegen falsch, weil&nbsp; $B$&nbsp; und&nbsp; $C$&nbsp; nicht disjunkt sind.
+*The third suggestion, on the other hand, is wrong because&nbsp; $B$&nbsp; and&nbsp; $C$&nbsp; are not disjoint.
 {{ML-Fuß}}

@@ Line 25: / Line 25: @@
 Hints:
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Mengentheoretische_Grundlagen|Mengentheoretische Grundlagen]].
+*The task belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Set_Theory_Basics|Set theory basics]].
-*Eine Zusammenfassung der theoretischen Grundlagen mit Beispielen bringt das Lernvideo [[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)|Mengentheoretische Begriffe und Gesetzmäßigkeiten]].
+*The topic of this chapter is illustrated with examples in the  (German language)  learning video[[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)|Mengentheoretische Begriffe und Gesetzmäßigkeiten]] \Rightarrow.

@@ Line 15: / Line 15: @@
 :$$G = (\overline A \cap \overline C) \cup (A \cap B \cap C).$$
-Überlegen Sie sich zunächst, welche Ziffern zu den Mengen&nbsp; $D$,&nbsp; $E$,&nbsp; $F$&nbsp; und&nbsp; $H$&nbsp; gehören und beantworten Sie dann die folgenden Fragen. <br>Begründen Sie Ihre Antworten mengentheoretisch.
+First consider which digits belong to the sets&nbsp; $D$,&nbsp; $E$,&nbsp; $F$&nbsp; and&nbsp; $H$&nbsp;  and then answer the following questions. <br>Justify your answers in terms of set theory.
@@ Line 24: / Line 24: @@
-''Hinweise:''
+Hints:
 *Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Mengentheoretische_Grundlagen|Mengentheoretische Grundlagen]].

← Older revision		Revision as of 15:47, 12 October 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^1.2 ~~Mengentheoretische Grundlagen~~	+	[[Category:Theory of Stochastic Signals: Exercises\|^1.2 Set Theory Basics
	^]]		^]]