160 Chapter 13

In the exercises, you will extend this theorem to higher dimensions. The real line R

is also equinumerous with three-dimensional space R

3

and, indeed, with R

n

in any finite

dimension and also with infinite dimensional space R

N

.

Mathematical Habits

Anthropomorphize. Understand a difficult idea by perceiving it from the perspective

of one of its parts, as though that part were a person, playing a certain role in a structure

or organization. Understand how the parts of the problem interact by imagining that

they are people who are fighting, when those parts are in tension, or cooperating, when

those parts support each other. Find a way to describe a mathematical idea in human

terms, in terms of human experience, not in order to make it relevant to humanity but,

rather, just to understand better the mathematical idea itself.

Use abstraction. Generalize your arguments via abstraction. Strip away the details

of an argument to find the core mathematical idea. Do not fear abstraction—master

abstraction by becoming deeply familiar with concrete instances.

Exercises

13.1 Suppose that guests arrived at Hilbert’s hotel in the manner described in the chapter: first

6 guests arrive one by one, and then 1000 all at once, and then Hilbert’s bus, followed by

Hilbert’s train, and finally Hilbert’s half marathon. If the manager followed the procedure

mentioned in the chapter, describe who are the occupants of rooms 0 through 100. How did

they arrive and with which party? In which car or seat were they when they arrived, if they

arrived by train or bus? Which fraction did they wear in the marathon? If you were the very

first guest to arrive, where do you end up in the end? And what is the first room above you

that is occupied? How did that guest arrive?

13.2 Down the street from Hilbert’s hotel is Hilbert’s co-op apartment complex, which is an infinite

cubical building, like N × N × N, where every occupant’s residence can be described by a

floor number n, a hallway number h, and a corridor r. Because the interior rooms have very

little light, the entire cooperative wants to move to Hilbert’s hotel. How can the manager

accommodate them?

13.3 Consider a dozen or so of the most familiar functions on the real numbers seen in a typical

calculus class. Which are functions from R to R? Which are injective? surjective? bijective?

13.4 Give examples of functions f : R → R that exhibit all possible combinations: bijective,

injective not surjective, surjective not injective, neither injective nor surjective.

13.5 Prove that if there is a surjective function f : N → A, then A is countable.