Mathematical Thinking

Due November 28

3:2; 1-8, 13-19, 28

Describe the set of natural numbers.*Au Natural.*

Describe each of the sets given below in words.*Au not-so-natural.*

{3, 6, 12, 15, 18, ….}

{1, 2, 3, 4, 5}

{1/2, 1/4, 1/8, 1/16, ….}

{-1, -2, -3, -4, ….}

{1, 4, 9, 16, 25, 36, 49, 64, 81, 100}

**Set setup.**We can denote the natural numbers symbolically as { 1, 2, 3, 4, …} Use this notation to express each of the sets described below.- The set of natural numbers less than 10.
- The set of all even natural numbers
- The set of solutions to the equation x² - 4 = 0
- The set of all reciprocals of the natural numbers.

**Little or large.**Which of the sets in Mindscape I.3 are infinite sets? Which are finite?

**A word you can count on.**Define the*cardinality*of the set.

**Even odds.**Let**E**stand for the set of all even natural numbers

(so **E** = {2,4,6,8,….}) and **O**
stand for the set of all odd natural numbers

(so **O** = {1,3,5,7,….}). Show
that the sets **E** and **O** have the same cardinality by describing an
explicit one-o-one correspondence between the two sets.

**Naturally even.**Let**E**stand for the set of all even natural numbers

(so **E** = {2,4,6,8,….}). Show
that the set **E** and the set of all natural numbers have the same
cardinality by describing an explicit one-to-one correspondence between the two
sets.

**Five takes over.**Let**EIF**be the set of al natural numbers ending in 5 (**EIF**stands

** **for “ends in five”).
That is, **EIF** = {5, 15, 25, 35, 45, 55, 65, 75,….}. Describe a

one-to-one correspondence
between the set of natural numbers and the set of **EIF.**

** **

13. **Squaring
off.** Let **S** stand for the set of all natural numbers that are
perfect squares, so **S** = {1, 4, 9, 16, 25, 36, 49, 64, ….}.
Show that the set **S **and the set of all natural numbers have the same
cardinality by describing an explicit one-to-one correspondence between the two
sets.

14. **Counting
cubes (formerly crows).** Let **C** stand for the set of all natural
numbers that are perfect cubes, so **C** = {1, 8, 27, 64, 125, 216, 343,
512, ….}. Show that the set **C** and the set of all natural numbers have
the same cardinality by describing an explicit one-to-one correspondence
between the two sets.

15. **Reciprocals.** Suppose **R
**is the set defined by

**R** = {1/1, 1/2, 1/3, 1/4,1/5,
1/6, ….}

Describe the set **R** in
words. Show that it has the same cardinality as the set of natural
numbers.

**Hotel Cardinality (formerly California) (H).**It is the stranded traveler’s fantasy. The Hotel Cardinality is a full-service luxury hotel with bar and restaurant. It has as many rooms as there are natural numbers. The room numbers are 1, 2, 3, 4, 5, ….You can see why stranded travelers love Hotel Cardinality. There appears to be no need for the sad sign: No Vacancy. Suppose, however, that every room is occupied. Now it appears that the night manager must flash the No Vacancy sign. What if a weary traveler were to arrive late at night looking for a place to stay? Could the night manager figure out a way to provide the traveler with a private room (no sharing) without evicting another guest? The answer is yes. Describe how this accommodation can be made; of course, some guests will have to move to other rooms.

**Hotel Cardinality continued.**Given the scenario in Mindscape II.16, suppose now that two more travelers arrive, each wanting his or her own private room. Is it possible for the night manager to make room for these folks without pushing anyone onto the streets?

**More Hotel C.**Given the scenario in Mindscape II.16—that is, the hotel starts full—suppose now that the Infinite Life Insurance Company, which has lots of employees—in fact, there are as many employees as there are natural numbers—decides to provide each of its employees with a private room. Is it possible for the night manager to give each of the infinitely many employees his or her own room without kicking any of the other guests onto the streets? By the way, as you may have guessed, after this busy evening, the night manager quit and got a job as the night manager of a nearby Motel 6 (it only had 56 rooms).

**So much sand.**Prove that there cannot be an infinite number of grains of sand on Earth.

28. **Bowling
ball barrel. **Suppose you have infinitely many bowling balls and two huge
barrels. You take the bowling balls and put each ball in one of the two barrels.
What can you conclude about the cardinality of at least one of the barrel?
Prove your answer.