13. All binary strings with is defined recursively as follows: Basis Step: ES Recursive Step: If ze S, then Or ES and 21 € S. where e is an empty string. Which of the following strings are not in S. (a) 0000 (b) 101 (c) 111 (d) 011 (e) 001

Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .