Please answer exercise 8. Exercise 8. Find the mistake in the following argument
ID: 3145311 • Letter: P
Question
Please answer exercise 8.
Exercise 8. Find the mistake in the following argument. (Hint: In the inductive step, an implicit assump- tion is made which is valid for most values of n but not for all. What is this implicit assumption and for which values of n is it valid?) We claim that all horses have the same color. It suffices to show that for each n e N, for each set of n horses, all of the horses i n the set have the same color. We shall prove this by induction on n. Let P(n) be the sentence For each set of n horses, all of the horses in the set have the same color BASE CASE: Clearly P(1) is true, because all of the horses in a set containing only one horse have the same color INDUCTIVE STEP: Let n e N such that P(n) is true. Consider any set of n + 1 horses. Removing one of the horses from the set, we obtain a set of n horses, all of which have the same color by the inductive hypothesis. Removing a different horse from the set of n +1 horses, we obtain another set of n hors all of these horses have the same color by the inductive hypothesis. Thus all of the horses in the set of n+1 horses have the same color. Hence P(n +1) is true too. CONCLUSION: Therefore by induction, for each n E N, P(n) is true. ' es andExplanation / Answer
The inductive step is valid for almost all n.For a large n,it is natural argument from the hypothesis.But the inductive step fails when we take n=2.When we have only two horses we can not conclude that the removed horse is the same colour because there left left no horse in the set to apply the transibility .So the inductive step it true for n=1 but fails for n=2.