We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Mathematical reasoning has to be pursued with great care, as there are pits that beset the unwary. We will begin with proofs by induction before going to other aspects of finite mathematics. Effecting a proof by induction is a sophisticated procedure that many students find quite mysterious. As long as they harbour the suspicion that somehow they are assuming what they have to prove, they are likely to treat it as a rote process and fall into confusion.
Perhaps the typographical error in the running head for page 589 of Michael Sullivan's College Algebra (Prentice Hall, 1995) says it all. Section 9.4, which treats mathematical induction, is headed Mathematical Indirection.
Rabbits reproduce; integers don't
Many readers will be familiar with some form of the following argument that all rabbits are the same color:
Clearly one rabbit has the same color. Suppose any set of n rabbits has the same color, say white, and consider a set of n + 1 rabbits. Remove one rabbit from the set; the induction hypothesis tells us that the remaining n rabbits are white. To see that the removed rabbit is also white, put it back in the set and remove some other rabbit, obtaining another set of n rabbits, which by the induction hypothesis must also be white. Thus all rabbits are white. ⋄
It is difficult to assess the mental demeanour of the upper division computer science student who provided this analysis:
The faulty logic in the rabbit problem has to do with the behavior of rabbits and integers.[…]
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
