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Homework Assignment 2

Assigned Tuesday September 17; due Friday September 27, 2PM at SS 1071

this document in PDF: HW02.pdf

Required reading

All of Spivak Chapters 2 and 3.

To be handed in

From Spivak Chapter 2: 1, 5.

From Spivak Chapter 3: 6, 13.

Recommended for extra practice

From Spivak Chapter 2: 3, 4, 12, 22.

From Spivak Chapter 3: 1, 7, 21.

An extra problem: Is there a problem with the following inductive proof that all horses are of the same color?

We assert that in all sets with precisely $ n$ horses, all horses are of the same color. For $ n=1$, this is obvious: it is clear that in a set with just one horse, all horses are of the same color. Now assume our assertion is true for all sets with $ n-1$ horses, and let us be given a set with $ n$ horses in it. By the inductive assumption, the first $ n-1$ of those are of the same color and also the last $ n-1$ of those. Hence they are all of the same color as illustrated below:

$\displaystyle \left(\begin{array}{c}
\end{array}\right] $

(The horses surrounded by round brackets $ (\cdots)$ are all of the same color. The horses surrounded by square brackets $ [\cdots]$ are all of the same color. Therefore the first and the last horses have the same color as the ones in the middle group, and hence all horses are of the same color.)

Just for fun

From Spivak Chapter 2: 27, 28.

A little more on Chapter 2, Problem 22:

Horse picture from http://lib.allconet.org/story_hour.htm.

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