Discrete Mathematics with Graph Theory (3rd Edition) 121

Discrete Mathematics with Graph Theory (3rd Edition) 121 -...

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Section 5.1 119 is true for n = k. Then k+l k ~)Xi - Xi-I) = (~)Xi - Xi-I)) + (Xk+l - X(k+l)-l) i=2 = (Xk - Xl) + (Xk+l - Xk) (by the induction hypothesis) which is the fonnula with n = k + 1. Thus, the result holds for all n ~ 2 by the Principle of Mathematical Induction. 11. [BB] The k to k + 1 step does not apply to the case k = 1. When k = 1, G = {al, a2}. Observe that the groups {al}, {a2} have no member in common. 12. [BB] The induction was not started properly. When n = 1, the left side is 1, while the right side is £. The statement is not true when n = 1. 13. As with the example at the end of this section, the problem is that the induction hypothesis has been applied to an integer to which it does not necessarily apply. The induction hypothesis applies only to integers e in the range 10 ~ e < k, 10 ~ e being implicit because all integers in this problem are at least no = 10. The induction hypothesis was applied to the integer k - 5, but this is not valid for k < 15. 14. The argument presented is flawed by its failure to specify and to verify the statement for n = no.
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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