Discrete Mathematics with Graph Theory (3rd Edition) 100

Discrete Mathematics with Graph Theory (3rd Edition) 100 -...

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98 Solutions to Exercises 7. (a) [BB] The given rule is w· a = 0 (mod 11) for a = (al,a2, ... , alO) and w = (1,1,1,1,1,1,1,1,1,1). (b) Suppose an error is made in the ith digit of an ISBN number so that instead of ala2 ... ai ... alO, the number appears as ala2' .. a~ ... alO, with a~ -:F ai. Then 0 = at + a2 ... + ai + ... + alO ¢ at + a2 + ... + a~ + ... + alO because [at + a2 + ... + ai + ... + alOl - [at + a2 + ... + a~ + ... + alOl = ai - a~ ¢ 0 (mod 11) since ai and a~ are digits between 0 and 9. (c) Suppose ata2 ... alO is a correct ISBN number. The transposition of ai and aHt cannot be detected because when we apply our test to the incorrect number at ... aH t ai ... alO, we compute at + ... + aiH + ai + ... + alO which is, of course, congruent to 0 (mod 11) because it's the exactly the same as at + ... + ai + aHt + ... + alO. 8. (a)[BB] 1-13579-02468-8; (d) [BB] 0-63042-00635-5; (b) 1-02468-13579-6; (e) 0-63000-16425-7; (c) 0-62608-39349-6; (0 8-29447-1 0493-1 . 9. (a) [BB] Not valid: 3(0 + 2 + 4 + 6 + 8 + 0) + (1 + 3 + 5 + 7 + 9 + 1) = 60 + 26 = 86 ¢ 0
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