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First Conundrum:
If possible, describe an induction principle for the set of
{0,−1,−2, . . .}. If this is not possible, give a brief explanation of why
it is not possible.
Second Conundrum:
Determine if there is a problem with the following proof by
induction that all Canadians live inWaterloo. Give a brief explanation
of the problem.
Base case: consider the group of 0 Canadians; clearly all the Canadians
in the group live in Waterloo.
I.H. For any group of 0 <= k Canadians, all those k Canadians live in
Waterloo.
Consider a group of k + 1 Canadians. Remove Canadian c from the
group. The group consists of k Canadians, by the I.H. all those k
Canadians live in Waterloo. Pick one of the k Canadians, say d, d
lives in Waterloo. Remove d from the group and replace d with c. The
group still has k Canadians so, by the I.H., they all live in Waterloo.
Therefore all the k + 1 Canadians live in Waterloo.
I'm not too sure I understand your notation for the solution to the first problem, could you restate that?
thanks for your patience
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