Question

Does the check digit of an ISSN detect every single error in an ISSN? Justify your answer with either a proof or a counterexample.

Short answer

Yes, the check digit of an ISSN detect every single error in an ISSN

Step-by-step solution

Step 1

For a International Standard Serial Number (ISSN) \({d_1}{d_2} \ldots {d_8}\), the check digit can be calculated using:

Note: When is used to represents \(d\_\left\{ 8 \right\}.\)

To proof: the check digit of a ISSN code detects all single errors

PROOF:

We assume that a single error implies that the value of a single digit changes.

Let the digit \({d_i}\)be the only digit that changes for some \(i^I \{ 1,2, \ldots ,7\} .\)

When \(i = 1\), then \(3{x_i}\)will then change by \(3,6,9,12,15,18,21, 24, or 27\) . Taking modulo 11 of these 9 changes in \(3{x_i}\),we obtain \(3,6,9,1,4,7,10,2, or 5\). Since the 9 possible changes all result in a distinct change modulo 11 different from 0 , the check digit will detect the error (as the congruence then no longer holds).

When \(i = 2\), then \(4{x_i}\)will then change by \(4,8,12,16,20,24,28, 32, or 36\). Taking modulo 11 of these 9 changes in \(4{x_i}\),we obtain \(4,8,1,5,9,2,6,10, or 3\). Since the 9 possible changes all result in a distinct change modulo 11 different from 0 , the check digit will detect the error (as the congruence then no longer holds).

When \(i = 3\), then \(5{x_i}\)will then change by \(5,10,15,20,25,30,35, 40, or 45\). Taking modulo 11 of these 9 changes in \(5{x_i}\),we obtain \(5,10,4,9,3,8,2,7, or 1\). Since the 9 possible changes all result in a distinct change modulo 11 different from 0 , the check digit will detect the error (as the congruence then no longer holds).

When \(i = 4,\)then \(6{x_i}\)will then change by \(6,12,18,24,30,36,42, 48, or 54\). Taking modulo 11 of these 9 changes in \(6{x_i}\), we obtain $\(6,1,7,2,8,3,9,4, or 10\). Since the 9 possible changes all result in a distinct change modulo 11 different from 0 , the check digit will detect the error (as the congruence then no longer holds).

Step 2

When \(i = 5\), then \(7{x_i}\)will then change by \(7,14,21,28,35,42,49,56, or 63\). Taking modulo 11 of these 9 changes in \(7{x_i}\),we obtain \(7,3,10,6,2,9,5,1, or 8\). Since the 9 possible changes all result in a distinct change modulo 11 different from 0 , the check digit will detect the error (as the congruence then no longer holds).

When \(i = 6\), then \(8 x\_\left\{ i \right\}\)will then change by \(8,16,24,32,40,48,56,64, or 72\). Taking modulo 11 of these 9 changes in \(8 x\_\left\{ i \right\},\)we obtain \(8,5,2,10,7,4,1,9, or 6\). Since the 9 possible changes all result in a distinct change modulo 11 different from 0 , the check digit will detect the error (as the congruence then no longer holds).

When \(i = 7,\)then \(9{x_i}\)will then change by \(9,18,27,36,45,54,63, 72, or 81\). Taking modulo 11 of these 9 changes in \(9{x_i}\), we obtain \(9,7,5,3,1,10,8,6, or 4\). Since the 9 possible changes all result in a distinct change modulo 11 different from 0 , the check digit will detect the error (as the congruence then no longer holds).

Thus we can then conclude that the check digit of a ISSN code detects all single errors.