Aufgaben:Exercise 1.1: For Labeling Books: Difference between revisions

Latest revision as of 17:53, 16 March 2026

Since the 1960s, all books are provided with a 10-digit "International Standard Book Number" $\rm (ISBN)$. The last digit of this so-called ISBN-10 specification is calculated according to the following rule:

$$ z_{10}= \left ( \sum_{i=1}^{9} \hspace{0.2cm} i \cdot z_i \right ) \hspace{-0.2cm} \mod 11 \hspace{0.05cm}.$$

Since 2007, the specification according to the standard ISBN-13 is additionally mandatory, whereby the check digit $z_{\rm 13}$ then results as follows:

$$z_{13} = 10 - \left ( \sum_{i=1}^{12} \hspace{0.2cm} z_i \cdot 3^{(i+1)\mod 2} \right ) \hspace{-0.2cm} \mod 10 \hspace{0.05cm}.$$

Some exemplary "ISBNs" are given opposite. The following questions refer to these.

Hints: This exercise belongs to the chapter "Objective of Channel Coding"

Questions

	ISBN-10,
	ISBN-13.

	Yes,
	No.

	Yes,
	No.

$M \ = \ $ $\ \rm$

	This is not a valid ISBN.
	The ISBN could be correct.
	The ISBN is certainly correct.

Solution

(1) Just by counting the ISBN digits, you can tell that answer 2 is correct. The weighted sum over all digits is a multiple of 10:

$$S \ = \ \hspace{-0.1cm} \sum_{i=1}^{13} \hspace{0.2cm} z_i \cdot 3^{(i+1) \hspace{-0.2cm} \cdot 2} = (9+8+8+7+6+8) \cdot 1 + (7+3+2+3+0+4) \cdot 3 = 110\hspace{0.3cm}\Rightarrow \hspace{0.3cm} S \hspace{-0.2cm} \mod 10 \hspace{0.15cm}\underline {= 0} \hspace{0.05cm}.$$

(2) The answer is No. Only one cancellation can be reconstructed with a single check digit.

(3) One digit can be reconstructed ⇒ Yes. For the digit $z_{\rm 8}$, it must hold:

$$[(9+8+4+3+0+1+2) \cdot 1 + (7+3+5+z_8+7+5) \cdot 3] \hspace{-0.2cm} \mod 10 = 0\hspace{0.3cm}\Rightarrow \hspace{0.3cm} [108 + 3z_8] \hspace{-0.2cm} \mod 10 = 0 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} z_8 \hspace{0.15cm}\underline {= 4} \hspace{0.05cm}.$$

(4) By the modulo 11 operation, $z_{10}$ can take the values $0,\ 1,\ \text{...} ,\ 10$ ⇒ $\underline{M =11}$.

Since "10" is not a digit, one makes do with $z_{10} = \rm X$.
This corresponds to the Roman representation of the number "10".

(5) The test condition is:

$$\ \ S= \left ( \sum_{i=1}^{10} \hspace{0.2cm} i \cdot z_i \right ) \hspace{-0.2cm} \mod 11 = 0 \hspace{0.05cm}.$$

The given ISBN satisfies this condition:

$$3 \cdot 1 + 8 \cdot 2 + 2 \cdot 3 + 7 \cdot 4 + 3 \cdot 5 + 7 \cdot 6 + 0 \cdot 7 + 6 \cdot 8 + 4 \cdot 9 + 7 \cdot 10 = 264\hspace{0.3cm}⇒\hspace{0.3cm} S= 264 \hspace{-0.3cm} \mod 11 = 0 \hspace{0.05cm}.$$

Correct is statement 2, since the check sum $S = 0$ could result even with more than one error.

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Kanalcodierung/Zielsetzung_der_Kanalcodierung
+{{quiz-Header|Buchseite=Channel_Coding/Objective_of_Channel_Coding
 }}
-[[File:P_ID2380__KC_A_1_1.png|right|frame|ISBN–10? Oder ISBN–13?]]
+[[File:EN_KC_A_1_1.png|right|frame|'''ISBN–10'''? Or '''ISBN–13'''?]]
-Seit den 1960er Jahren werden alle Bücher mit einer 10–stelligen ''International Standard Book Number'' versehen. Die letzte Ziffer dieser sog. '''ISBN–10–Angabe''' berechnet sich dabei entsprechend folgender Regel:
+Since the 1960s,&nbsp; all books are provided with a 10-digit&nbsp; "International Standard Book Number"&nbsp; $\rm (ISBN)$.&nbsp; The last digit of this so-called&nbsp; '''ISBN-10 specification'''&nbsp; is calculated according to the following rule:
 :$$ z_{10}= \left ( \sum_{i=1}^{9} \hspace{0.2cm} i \cdot z_i \right ) \hspace{-0.2cm} \mod 11 \hspace{0.05cm}.$$
-Seit 2007 ist zusätzlich die Angabe entsprechend des Standards '''ISBN–13''' verpflichtend, wobei die Prüfziffer $z_{\rm 13}$ sich dann wie folgt ergibt:
+Since 2007,&nbsp; the specification according to the standard&nbsp; '''ISBN-13'''&nbsp; is additionally mandatory,&nbsp; whereby the check digit&nbsp; $z_{\rm 13}$&nbsp; then results as follows:
 :$$z_{13} = 10 - \left ( \sum_{i=1}^{12} \hspace{0.2cm} z_i \cdot 3^{(i+1)\mod 2} \right ) \hspace{-0.2cm} \mod 10 \hspace{0.05cm}.$$
-Nebenstehend sind einige beispielhafte ISBN angegeben. Hierauf beziehen sich die folgenden Fragen.
-''Hinweis'': Die Aufgabe gehört zum Themengebiet von Kapitel [[Kanalcodierung/Zielsetzung_der_Kanalcodierung|Zielsetzung_der_Kanalcodierung]]
+Some exemplary&nbsp; "ISBNs"&nbsp; are given opposite.&nbsp; The following questions refer to these.
-===Fragebogen===
-<quiz display=simple>
-Um welchen Standard handelt es sich bei Beispiel 1?
-|type="[]"}
-- ISBN–10,
-+ISBN–13.
-{Entsprechend Beispiel 2 sind zwei Ziffern einer ISBN–13 ausgelöscht. Kann man die ISBN rekonstruieren? Wenn Ja: Geben Sie die ISBN–13 an.
+Hints:&nbsp; This exercise belongs to the chapter&nbsp; [[Channel_Coding/Objective_of_Channel_Coding|"Objective of Channel Coding"]]
-|type="[]"}
-- ja,
-+Nein.
+===Questions===
-{Entsprechend Beispiel 3 ist eine Ziffer einer ISBN–13 ausgelöscht. Kann die ISBN rekonstruiert werden? Wenn Ja: Geben Sie die ISBN–13 an.
+<quiz display=simple>
-|type="[]"}
-+Ja,
--Nein.
-{Wieviele verschiedene Werte kann die Prüfziffer  $z_{\rm 10}$ bei ISBN–10 annehmen?
-|type="{}"}
-$M \ = \ $ { 11 3% } $\  \rm$
-{Mitgeteilt als ISBN–10 wird 3–8273–7064–7. Welche Aussage trifft zu?
+What is the standard for &nbsp;$\text{Example 1}$?
-|type="{}"}
+|type="()"}
+- ISBN-10,
++ISBN-13.
--Dies ist keine zulässige ISBN.
+{Accordingly &nbsp;$\text{Example 2}$&nbsp; two digits of an ISBN-13 are deleted.&nbsp; Is it possible to reconstruct the ISBN?&nbsp; If yes: &nbsp; Specify the ISBN-13.
-+Die ISBN könnte richtig sein.
+|type="()"}
--Die ISBN ist mit Sicherheit richtig.
+- Yes,
++No.
+{According to &nbsp;$\text{Example 3}$&nbsp; one digit of an ISBN-13 is erased.&nbsp; Can the ISBN be reconstructed?&nbsp; If Yes: &nbsp; Specify the ISBN-13.
+|type="()"}
++Yes,
+-No.
+{How many different values &nbsp;$(M)$&nbsp; can the check digit&nbsp; $z_{\rm 10}$&nbsp; take for ISBN-10?
+|type="{}"}
+$M \ = \ $ { 11 3% } $\ \rm$
+{Assumed as ISBN-10 is&nbsp; "3-8273-7064-7".&nbsp; Which statement is true?
+|type="()"}
+- This is not a valid ISBN.
++ The ISBN could be correct.
+- The ISBN is certainly correct.
+</quiz>
+===Solution===
+{{ML-Kopf}}
+'''(1)'''&nbsp; Just by counting the ISBN digits,&nbsp; you can tell that&nbsp; <u>answer 2</u>&nbsp; is correct.&nbsp; The weighted sum over all digits is a multiple of 10:
+:$$S \ = \ \hspace{-0.1cm} \sum_{i=1}^{13} \hspace{0.2cm} z_i \cdot 3^{(i+1) \hspace{-0.2cm} \cdot 2} = (9+8+8+7+6+8) \cdot 1 + (7+3+2+3+0+4) \cdot 3 = 110\hspace{0.3cm}\Rightarrow \hspace{0.3cm} S \hspace{-0.2cm} \mod 10 \hspace{0.15cm}\underline {= 0} \hspace{0.05cm}.$$
+'''(2)'''&nbsp; The answer is&nbsp; <u>No</u>.&nbsp; Only one cancellation can be reconstructed with a single check digit.
-{Input-Box Frage
-|type="{}"}
-$\alpha$ = { 0.3 }
+'''(3)'''&nbsp; One digit can be reconstructed &nbsp;⇒ &nbsp; <u>Yes</u>.&nbsp; For the digit&nbsp; $z_{\rm 8}$,&nbsp; it must hold:
+:$$[(9+8+4+3+0+1+2) \cdot 1 + (7+3+5+z_8+7+5) \cdot 3] \hspace{-0.2cm} \mod 10 = 0\hspace{0.3cm}\Rightarrow \hspace{0.3cm} [108 + 3z_8] \hspace{-0.2cm} \mod 10 = 0 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} z_8 \hspace{0.15cm}\underline {= 4} \hspace{0.05cm}.$$
+'''(4)'''&nbsp; By the modulo 11 operation,&nbsp; $z_{10}$&nbsp; can take the values&nbsp; $0,\ 1,\ \text{...} ,\ 10$ &nbsp; ⇒ &nbsp; $\underline{M =11}$.
+*Since&nbsp; "10"&nbsp; is not a digit,&nbsp; one makes do with&nbsp; $z_{10} = \rm X$.
+*This corresponds to the Roman representation of the number&nbsp; "10".
-</quiz>
-===Musterlösung===
+'''(5)'''&nbsp; The test condition is:
-{{ML-Kopf}}
+:$$\ \ S= \left ( \sum_{i=1}^{10} \hspace{0.2cm} i \cdot z_i \right ) \hspace{-0.2cm} \mod 11 = 0 \hspace{0.05cm}.$$
-'''1.'''
-'''2.'''
+*The given ISBN satisfies this condition:
-'''3.'''
+:$$3 \cdot 1 + 8 \cdot 2 + 2 \cdot 3 + 7 \cdot 4 + 3 \cdot 5 + 7 \cdot 6 + 0 \cdot 7 + 6 \cdot 8 + 4 \cdot 9 + 7 \cdot 10 = 264\hspace{0.3cm}⇒\hspace{0.3cm} S= 264 \hspace{-0.3cm} \mod 11 = 0 \hspace{0.05cm}.$$
-'''4.'''
+*Correct is&nbsp; <u>statement 2</u>,&nbsp; since the check sum&nbsp; $S = 0$&nbsp; could result even with more than one error.
-'''5.'''
-'''6.'''
-'''7.'''
 {{ML-Fuß}}
-[[Category:Aufgaben zu  Kanalcodierung|^1.1 Zielsetzung der Kanalcodierung
+[[Category:Channel Coding: Exercises|^1.1 Objective of Channel Coding^]]
+[[de:Aufgaben:Aufgabe 1.1: Zur Kennzeichnung aller Bücher]]
-^]]