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), the check digit can be calculated using:

Note: Whenis used to represents

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 digitbe the only digit that changes for some

When, thenwill then change by. Taking modulo 11 of these 9 changes in,we obtain. 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 congruencethen no longer holds).

When, thenwill then change by. Taking modulo 11 of these 9 changes in,we obtain. 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 congruencethen no longer holds).

When, thenwill then change by. Taking modulo 11 of these 9 changes in,we obtain. 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 congruencethen no longer holds).

Whenthenwill then change by. Taking modulo 11 of these 9 changes in,we obtain $. 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 congruencethen no longer holds).

Step 2

When, thenwill then change by. Taking modulo 11 of these 9 changes in,we obtain. 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 congruencethen no longer holds).

When, thenwill then change by. Taking modulo 11 of these 9 changes inwe obtain. 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 congruencethen no longer holds).

Whenthenwill then change by. Taking modulo 11 of these 9 changes in, we obtain. 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 congruencethen no longer holds).

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