Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:

(16)

Next: Wang Ying's Solution of Homework Assignment 2 Previous: Class Notes for the Week of September 9 (8 of 8)

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:

$$\left(\begin{array}{c} \includegraphics[width=0.6in]{RockingHorse... ...n{array}{c} \includegraphics[width=0.6in]{RockingHorse.eps} \end{array}\right]$$

(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:

• We know that if $a$ and $b$ are non-negative then $\sqrt{ab}\leq(a+b)/2$. This is the same as saying that $4ab\leq(a+b)^2$, which is the same as saying that the area of four $a$ by $b$ rectangles is less than or equal to the area of a square with side $a+b$. Can you actually fit four $a$ by $b$ rectangles inside a square of side $a+b$ without overlaps? It's fun and not too hard.

• We know that if $a$, $b$ and $c$ are non-negative then $\sqrt[3]{abc}\leq(a+b+c)/3$. This is the same as saying that $27abc\leq(a+b+c)^3$, which is the same as saying that the volume of 27 $a$ by $b$ by $c$ rectangular boxes is less than or equal to the volume of a cube with side $a+b+c$. Can you actually fit 27 such $a$ by $b$ by $c$ rectangular boxes inside a cube of side $a+b+c$ without overlaps? This is also fun, but quite hard. You have no chance of doing it without a physical model. Make yourself one!

• The corresponding problem in 4D, involving 256 boxes of size $a\times b\times c\times d$, is actually a little easier, though trickier, than the 3D problem. Can you do it?

• The corresponding problem in 5D, involving 3,125 boxes of size $a\times b\times c\times d\times e$, is an open problem -- meaning that nobody knows how to solve it. Can you?

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

About this document ...

This document was generated using the LaTeX2HTML translator Version 2K.1beta (1.49)

Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.

The command line arguments were:
latex2html -no_math -html_version 3.2 -dir l2h -split 0 -long_titles 4 -show_section_numbers -no_navigation -no_auto_link -white -no_footnode HW02.tex

The translation was initiated by Dror Bar-Natan on 2002-09-30