Aufgaben:Exercise 1.2: Decimal/Binary Converter - Revision history

Noah at 19:33, 28 November 2021

2021-11-28T19:33:20Z

Noah at 15:12, 23 November 2021

2021-11-23T15:12:00Z

Guenter at 13:06, 19 November 2021

2021-11-19T13:06:24Z

Guenter at 17:18, 18 November 2021

2021-11-18T17:18:15Z

Guenter: Guenter moved page Aufgaben:Exercise 1.2: Switching Logic (D/B Converter) to Aufgaben:Exercise 1.2: Decimal/Binary Converter

2021-11-18T17:06:31Z

Guenter moved page Exercise 1.2: Switching Logic (D/B Converter) to Exercise 1.2: Decimal/Binary Converter

Guenter at 17:03, 18 November 2021

2021-11-18T17:03:34Z

Noah at 21:59, 16 November 2021

2021-11-16T21:59:05Z

Noah at 21:58, 16 November 2021

2021-11-16T21:58:45Z

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Guenter at 15:46, 12 October 2021

2021-10-12T15:46:09Z

Guenter: Guenter moved page Aufgaben:Aufgabe 1.2: Schaltlogik (D/B-Wandler) to Aufgaben:Exercise 1.2: Switching Logic (D/B Converter)

2021-10-12T15:41:22Z

Guenter moved page Aufgabe 1.2: Schaltlogik (D/B-Wandler) to Exercise 1.2: Switching Logic (D/B Converter)

@@ Line 24: / Line 24: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Mengentheoretische_Grundlagen|Set theory basic]].
-*The topic of this chapter is illustrated with examples can be found in the (German language) learning video:&nbsp;
+*The topic of this chapter is illustrated with examples in the (German language) learning video:&nbsp;
 :[[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)|"Mengentheoretische Begriffe und Gesetzmäßigkeiten"]] &nbsp; &rArr; &nbsp; "Set-theoretical terms and laws".

← Older revision		Revision as of 15:12, 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_A_1_2.png\|right\|frame\|Logical circuit for D/B converting]]		[[File:EN_Sto_A_1_2.png\|right\|frame\|Logical circuit for D/B converting]]

@@ Line 72: / Line 72: @@
-⇒  &nbsp;<u>Proposed solutions 2 and 4</u> are correct.
+⇒  &nbsp;<u>Proposed solutions 2 and 4</u>&nbsp; are correct.
-'''(2)'''&nbsp; The event&nbsp; $V$&nbsp; consists of the two numbers&nbsp; $4$&nbsp; (binary 0100) and&nbsp; $6$&nbsp; (binary 0110)   &nbsp;
+'''(2)'''&nbsp; The event&nbsp; $V$&nbsp; consists of the two numbers&nbsp; $4$&nbsp; (binary 0100) and&nbsp; $6$&nbsp; (binary 0110)   &nbsp; ⇒  &nbsp;  The correct <u>solutions are 1 and 3</u>.
-[[File:P_ID2848__Sto_A_1_2c.png|right|frame|Auxiliary Venn-diagram]]
+[[File:P_ID2848__Sto_A_1_2c.png|right|frame|Auxiliary Venn diagram]]
-'''(3)'''&nbsp; For the event&nbsp; $W$&nbsp;, de Morgan's theorem holds:
+'''(3)'''&nbsp; For the event&nbsp; $W$,&nbsp; de Morgan's theorem holds:
 :$$\overline W = \overline A \cup \overline D \cup (\overline B \cap C) \cup (B \cap \overline C)
 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} W = \overline{\overline W} = A \cap D \cap (\overline{\overline B \cap C}) \cap (\overline{B \cap \overline C}).$$
-*Using de Morgan's theorems, it further follows:
+*Using de Morgan's theorems,&nbsp; it further follows:
 :$$ W = A \cap D \cap (B \cup \overline C) \cap (\overline B \cup C).$$
-*Finally, using the Boolean relation&nbsp; $(B \cup \overline C) \cap (\overline B \cup C) = (B \cap C) \cup (\overline B \cap \overline C)$&nbsp; we obtain (see sketch):
+*Finally,&nbsp; using the Boolean relation&nbsp; $(B \cup \overline C) \cap (\overline B \cup C) = (B \cap C) \cup (\overline B \cap \overline C)$&nbsp; we obtain&nbsp; (see sketch):
 :$$W = (A \cap B \cap C \cap D) \cup (A \cap \overline B \cap \overline C \cap D).$$
-*Thus,&nbsp; $W$&nbsp; contains the numbers&nbsp; $15$&nbsp; and&nbsp; $9$&nbsp;  &nbsp;
+*Thus,&nbsp; $W$&nbsp; contains the numbers&nbsp; $15$&nbsp; and&nbsp; $9$&nbsp;  &nbsp; ⇒  &nbsp;  only the&nbsp; <u>proposed solution 1</u>&nbsp; is correct.
 '''(4)'''&nbsp; The union of&nbsp; $U$,&nbsp; $V$&nbsp; and&nbsp; $W$&nbsp; contains the following numbers: &nbsp; $4, 6, 8, 9, 10, 12, 14, 15$.
-*Accordingly, the set&nbsp; $P$&nbsp; as the complement of this union is: &nbsp; $P \in {\{1, 2, 3, 5, 7, 11, 13\}}$.
+*Accordingly, the set&nbsp; $P$&nbsp; as the complement of this union is: &nbsp;
 *These are exactly the prime numbers which can be represented with four bits  &nbsp; ⇒  &nbsp;  <u>Proposed solution 2</u>.

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Stochastische Signaltheorie/Mengentheoretische Grundlagen}}
-[[File:EN_Sto_A_1_2.png|right|frame|Logisches Schaltwerk (Sequential logic unit)???]]
+[[File:EN_Sto_A_1_2.png|right|frame|Logical circuit for D/B converting]]
 A number generator&nbsp; $Z$&nbsp; supplies decimal values in the range&nbsp; $1$&nbsp; to&nbsp; $15$.
 *These are converted into binary numbers&nbsp; (block outlined in red).
@@ Line 8: / Line 8: @@
 *For example&nbsp; $Z = 11$&nbsp; delivers the binary values
 :$$ A = 1, \ B = 0, \ C = 1, \ D = 1. $$
-Set theoretically, this can be represented as follows:
+Set-theoretically,&nbsp; this can be represented as follows:
 :$$ Z = 11\qquad\widehat{=}\qquad A \cap\overline{ B} \cap C \cap D.$$
@@ Line 16: / Line 16: @@
 :$$W,\;  {\rm where} \; \, \overline{W} = \overline{A} \cup \overline{D} \cup (\overline{B} \cap C) \cup (B \cap \overline{C}). $$
 *Note that&nbsp; $Z = 0 \ ⇒ \ A = B = C = D = 0$&nbsp; is already excluded by the number generator.
-*Also note that not all input quantities&nbsp; $A$,&nbsp; $B$,&nbsp; $C$&nbsp; and&nbsp; $D$&nbsp; are used to calculate all intermediate quantities&nbsp; $U$,&nbsp; $V$&nbsp; and&nbsp; $W$&nbsp; respectively.
+*Note also that not all input quantities&nbsp; $A$,&nbsp; $B$,&nbsp; $C$&nbsp; and&nbsp; $D$&nbsp; are used to calculate all intermediate quantities&nbsp; $U$,&nbsp; $V$&nbsp; and&nbsp; $W$,&nbsp; resp.
@@ Line 22: / Line 22: @@
+:[[Mengentheoretische_Begriffe_und_Gesetzmäßigkeiten_(Lernvideo)|"Mengentheoretische Begriffe und Gesetzmäßigkeiten"]] &nbsp; &rArr; &nbsp; "Set-theoretical terms and laws".
@@ Line 35: / Line 34: @@
 <quiz display=simple>
-{Which statements are true regarding the random variable&nbsp; $U$&nbsp; zutreffend?
+{Which statements are true regarding the random variable&nbsp; $U$?
 |type="[]"}
 - $U$&nbsp; contains two elements.
@@ Line 42: / Line 41: @@
 + The largest element of&nbsp; $U$&nbsp; is&nbsp; $14$.
-{Which statements are true regarding the random variable&nbsp; $V$&nbsp;?
+{Which statements are true regarding the random variable&nbsp; $V$?
 |type="[]"}
 + $V$&nbsp; contains two elements.
@@ Line 49: / Line 48: @@
 - The largest element of&nbsp; $V$&nbsp; is&nbsp; $14$.
-{Which statements are true regarding the random variable&nbsp; $W$&nbsp;?
+{Which statements are true regarding the random variable&nbsp; $W$?
 |type="[]"}
 + $W$&nbsp; contains two elements.
@@ Line 60: / Line 59: @@
 - $P$&nbsp; contains all powers of two.
 + $P$&nbsp; contains all prime numbers.
-- $P$&nbsp; describes the empty set <math>\phi</math> .
+- $P$&nbsp; describes the empty set&nbsp; $\phi$.
 - $P$&nbsp; is identical with the universal set&nbsp; $G = {1,2, \ \text{...} \ , 15}$.

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Stochastische Signaltheorie/Mengentheoretische Grundlagen}}
-==A1.2 Switching logic (D/B converter)==
 [[File:EN_Sto_A_1_2.png|right|frame|Logisches Schaltwerk (Sequential logic unit)???]]
 A number generator&nbsp; $Z$&nbsp; supplies decimal values in the range&nbsp; $1$&nbsp; to&nbsp; $15$.

← Older revision	Revision as of 17:06, 18 November 2021
(No difference)

← Older revision		Revision as of 15:46, 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^]]

← Older revision	Revision as of 15:41, 12 October 2021
(No difference)