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4 - Finite Mathematics

Edward J. Barbeau
Edward J. Barbeau
Affiliation:
University of Toronto
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Summary

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.[…]

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Mathematical Fallacies, Flaws, and Flimflam , pp. 63 - 76
Publisher: Mathematical Association of America
Print publication year: 2000

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  • Finite Mathematics
  • Edward J. Barbeau, University of Toronto
  • Book: Mathematical Fallacies, Flaws, and Flimflam
  • Online publication: 05 December 2013
  • Chapter DOI: https://doi.org/10.5948/UPO9781614445180.006
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  • Finite Mathematics
  • Edward J. Barbeau, University of Toronto
  • Book: Mathematical Fallacies, Flaws, and Flimflam
  • Online publication: 05 December 2013
  • Chapter DOI: https://doi.org/10.5948/UPO9781614445180.006
Available formats
×

Save book to Google Drive

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.

  • Finite Mathematics
  • Edward J. Barbeau, University of Toronto
  • Book: Mathematical Fallacies, Flaws, and Flimflam
  • Online publication: 05 December 2013
  • Chapter DOI: https://doi.org/10.5948/UPO9781614445180.006
Available formats
×