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182
chapter_7/exercises.md
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@ -3932,3 +3932,185 @@ Omitted.
Justify your answer.
Omitted.
---
Page 507
**Exercise Set 7.4**
1. When asked what it means to say that set $A$ has the same cardinality as set
$B$, a student replies, "$A$ and $B$ are one-to-one and onto." What _should_
the student have replied? Why?
2. Show that "there are as many squares as there are numbers" by exhibiting a
one-to-one correspondence from the positive integers, $\mathbb{Z}^+$, to the
set $S$ of all squares of positive integers:
$$ S = \{n \in \mathbb{Z}^+ | n = k^2, \text{ for some positive integer } k\} $$
3. Let
$3\mathbb{Z} = \{n \in \mathbb{Z} | n = 3k, \text{ for some integer } k\}$.
Prove that $\mathbb{Z}$ and $3\mathbb{Z}$ have the same cardinality.
4. Let $\mathbb{O}$ be the set of all odd integers. Prove that $\mathbb{O}$ has
the same cardinality as $2\mathbb{Z}$, the set of all even integers.
5. Let $25\mathbb{Z}$ be the set of all integers that are multiples of $25$.
Prove that $25\mathbb{Z}$ has the same cardinality as $2\mathbb{Z}$, the set
of all even integers.
6. Use the functions $I$ and $J$ defined in the paragraph following Example
7.4.1 to show that even though there is a one-to-one correspondence, $H$,
from $2\mathbb{Z}$ to $\mathbb{Z}$, there is also a function from
$2\mathbb{Z}$ to $\mathbb{Z}$ that is one-to-one but not onto and a function
from $\mathbb{Z}$ to $2\mathbb{Z}$ that is onto but not one-to-one. In other
words, show that $I$ is one-to-one but not onto, and show that $J$ is onto
but not one-to-one.
7.
a. Check that the formula for $F$ given at the end of Example 7.4.2 produces the
correct values for $n = 1, 2, 3, \text{ and } 4$.
b. Use the floor function to write a formula for $F$ as a single algebraic
expression for each positive integer $n$.
8. Use the result of exercise 3 to prove that $3\mathbb{Z}$ is countable.
9. Show that the set of all nonnegative integers is countable by exhibiting a
one-to-one correspondence between $\mathbb{Z}^+$ and
$\mathbb{Z}^{\text{nonneg}}$.
In 10-14 $S$ denotes the set of real numbers strictly between $0$ and $1$. That
is, $s = \{x \in \mathbb{R} | 0 < x < 1\}$.
10. Let $U = \{x \in \mathbb{R} | 0 < x < 2\}$. Prove that $S$ and $U$ have the
same cardinality.
11. Let $V = \{x \in \mathbb{R} | 2 < x < 5\}$. Prove that $S$ and $V$ have the
same cardinality.
12. Let $a$ and $b$ be real numbers with $a < b$, and suppose that
$W = \{x \in \mathbb{R} | a < x < b\}$. Prove that $S$ and $W$ have the same
cardinality.
13. Draw the graph of the function $f$ defined by the following formula:
For each real number $x$ with $0 < x < 1$,
$$ f(x) = \tan\left(\pi x - \frac{\pi}{2}\right) $$
Use the graph to explain why $S$ and $\mathbb{R}$ have the same cardinality.
14. Define a function $g$ from the set of real numbers to $S$ by the following
formula:
For each real number $x$,
$$ g(x) = \frac{1}{2} \cdot \left(\frac{x}{1 + |x|}\right) + \frac{1}{2} $$
Prove that $g$ is a one-to-one correspondence. (It is possible to prove this
statement either with calculus or without it.) What conclusion can you draw from
this fact?
15. Show that the set of all bit strings (strings of $0$'s and $1$'s) is
countable.
16. Show that $\mathbb{Q}$, the set of all rational numbers, is countable.
17. Show that $\mathbb{Q}$, the set of all rational numbers, is dense along the
number line by showing that given any two rational numbers $r_1$ and $r_2$
with $r_2 < r_2$, there exists a rational number $x$ such that
$r_1 < x < r_2$.
18. Must the average of two irrational numbers always be irrational? Prove or
give a counterexample.
19. Show that the set of all irrational numbers is dense along the number line
by showing that given any two real numbers, there is an irrational number in
between.
20. Give two examples of functions from $\mathbb{Z}$ to $\mathbb{Z}$ that are
one-to-one but not onto.
21. Give two examples of functions from $\mathbb{Z}$ to $\mathbb{Z}$ that are
onto but not one-to-one.
22. Define a function: $g: \mathbb{Z}^+ \times \mathbb{Z}y+ \to \mathbb{Z}^+$ by
the formula $g(m, n) = 2^m3^n$ for all
$(m, n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$. Show that $g$ is one-to-one
and use this result to prove that $\mathbb{Z}^+ \times \mathbb{Z}^+$ is
countable.
23.
a. Explain how to use the following diagram to show that
$\mathbb{Z}^{\text{nonneg}} \times \mathbb{Z}^{\text{nonneg}}$ and
$\mathbb{Z}^{\text{nonneg}}$ have the same cardinality.
(See Page 508 for image.)
b. Define a function
$H: \mathbb{Z}^{\text{nonneg}} \times \mathbb{Z}^{\text{nonneg}} \to \mathbb{Z}^{\text{nonneg}}$
by the formula
$$ H(m, n) = n + \frac{(m + n)(m + n + 1)}{2} $$
for all nonnegative integers $m$ and $n$. Interpret the action of $H$
geometrically using the diagram of part (a).
24. Prove that the function $H$ defined analytically in exercise 23b is a
one-to-one correspondence.
25. Prove that $0.1999 \dots = 0.2$.
26. Prove that any infinite set contains a countably infinite subset.
27. Prove that if $A$ is any countably infinite set, $B$ is any set, and
$g: A \to B$ is onto, then $B$ is countable.
28. Prove that a disjoint union of any finite set and any countably infinite set
is countably infinite.
29. Prove that a union of any two countably infinite sets is countably infinite.
30. Use the result of exercise 29 to prove that the set of all irrational
numbers is uncountable.
31. Use the results of exercises 28 and 29 to prove that a union of any two
countable sets is countable.
32. Prove that $\mathbb{Z} \times \mathbb{Z}$, the Cartesian product of the set
of integers with itself, is countably infinite.
33. Use the results of exercises 27, 31, and 32 to prove the following: If $R$
is the set of all solutions to all equations of the form $x^2 + bx + c = 0$,
where $b$ and $c$ are integers, then $R$ is countable.
34. Let $\mathscr{P}(S)$ be the set of all subsets of set $S$, and let $T$ be
the set of all functions from $S$ to $\{0, 1\}$. Show that $\mathscr{P}(S)$
and $T$ have the same cardinality.
35. Let $S$ be a set and let $\mathscr{P}(S)$ be the set of all subsets of $S$.
Show that $S$ is "smaller than" $\mathscr{P}(S)$ in the sense that there is
a one-to-one function from $S$ to $\mathscr{P}(S)$ but there is no onto
function from $S$ to $\mathscr{P}(S)$.
36. The Schroeder-Bernstein theorem states the following: if $A$ and $B$ are any
sets with the property that there is a one-to-one function from $A$ to $B$
and a one-to-one function from $B$ to $A$, then $A$ and $B$ have the same
cardinality. Use this theorem to prove that there are as many functions from
$\mathbb{Z}^+$ to $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ as there functions from
$\mathbb{Z}^+$ to $\{0, 1\}$.
37. Prove that if $A$ and $B$ are any countably infinite sets, then $A \times B$
is countably infinite.
38. Suppose $A_1, A_2, A_3, \dots$ is an infinite sequence of countable sets.
Recall that
$$ \bigcup_{i = 1}^{\infty}A_i = \{x | x \in A_i \text{ for some positive integer } i\} $$
Prove that $\bigcup_{i = 1}^{\infty}A_i$ is countable. (In other words, prove
that a countably infinite union of countable sets is countable.)

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chapter_7/notes.md
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@ -413,3 +413,218 @@ in $X$ such that
$$ (g \circ f)(x) = g(f(x)) = g(y) = z $$
_[as was to be shown]._ It follows that $g \circ f$ is onto.
---
Page 496
**Definition**
Let $A$ and $B$ be any sets. **$A$ has the same cardinality as $B$** if, and
only if, there is a one-to-one correspondence from $A$ to $B$. In other words,
$A$ has the same cardinality as $B$ if, and only if, there is a function $f$
from $A$ to $B$ that is one-to-one and onto.
---
**Theorem 7.4.1 Properties of Cardinality**
For all sets $A$, $B$, and $C$:
a. **Reflexive property of cardinality:** $A$ has the same cardinality as $A$.
b. **Symmetric property of cardinality:** If $A$ has the same cardinality as
$B$, then $B$ has the same cardinality as $A$.
c. **Transitive property of cardinality:** If $A$ has the same cardinality as
$B$ and $B$ has the same cardinality as $C$, then $A$ has the same cardinality
as $C$.
**Proof:**
_Part (a), Reflexivity:_
Suppose $A$ is any set. _[To show that $A$ has the same cardinality as $A$, we
must show there is a one-to-one correspondence from $A$ to $A$.]_ Consider the
identity function $I_A$ from $A$ to $A$. This function is one-to-one because if
$x_1$ and $x_2$ are any elements in $A$ with $I_A(x_1) = I_A(x_2)$, then, by
definition of $I_A$, $x_1 = x_2$. The identity function is also onto because if
$y$ is any element of $A$, then $y = I_A(y)$ by definition of $I_A$. Hence $I_A$
is a one-to-one correspondence from $A$ to $A$. _[So there exists a one-to-one
correspondence from $A$ to $A$, as was to be shown.]_
_Part (b), Symmetry:_
Suppose $A$ and $B$ are any sets and $A$ has the same cardinality as $B$. _[We
must show that $B$ has the same cardinality as $A$.]_ Since $A$ has the same
cardinality as $B$, there is a function $f$ from $A$ to $B$ that is one-to-one
and onto. But then, by Theorems 7.2.2 and 7.2.3, there is a function $f^{-1}$
from $B$ to $A$ that is also one-to-one and onto. Hence $B$ has the same
cardinality as $A$ _[as was to be shown]._
_Part \(c\), Transitivity:_
Suppose $A$, $B$, and $C$ are any sets and $A$ has the same cardinality as $B$
and $B$ has the same cardinality as $C$. _[We must show that $A$ has the same
cardinality as $C$.]_ Since $A$ has the same cardinality as $B$, there is a
function $f$ from $A$ to $B$ that is one-to-one and onto, and since $B$ has the
same cardinality as $C$, there is a function $g$ from $B$ to $C$ that is
one-to-one and onto. But then, by Theorems 7.3.3 and 7.3.4, $g \circ f$ is a
function from $A$ to $C$ that is one-to-one and onto. Hence $A$ has the same
cardinality as $C$ _[as was to be shown]._
---
Page 497
**Definition**
$A$ and $B$ **have the same cardinality** if, and only if, $A$ has the same
cardinality as $B$ or $B$ has the same cardinality as $A$.
---
Page 499
**Definition**
A set is **finite** if, and only if, it is the empty set or can be put into
one-to-one correspondence with a set of the form $\{1, 2, \dots, n\}$ for some
positive integer $n$. A set is **countably infinite** if, and only if, it has
the same cardinality as the set of positive integers $\mathbb{Z}^+$. A set is
**countable** if, and only if, it is finite or countably infinite. A set that is
not countable is called **uncountable**.
---
Page 502
**Theorem 7.4.2 (Cantor)**
The set of all real numbers between $0$ and $1$ is uncountable.
**Proof (by contradiction):**
Suppose the set of all real numbers between $0$ and $1$ is countable. Then the
decimal representations of these numbers can be written in a list as follows:
$$ 0.a_{11}a_{12}a_{13}\cdots a_{1n}\cdots $$
$$ 0.a_{21}a_{22}a_{23}\cdots a_{2n}\cdots $$
$$ 0.a_{31}a_{32}a_{33}\cdots a_{3n}\cdots $$
$$ \vdots $$
$$ 0.a_{n1}a_{n2}a_{n3}\cdots a_{nn}\cdots $$
$$ \vdots $$
_[We will derive a contradiction by showing that there is a number between $0$
and $1$ that does not appear on this list.]_
For each pair of positive integers $i$ and $j$, the $j$th decimal digit of the
$i$th number on the list is $a_{ij}$. In particular, the first decimal digit of
the first number on the list is $a_{11}$, the second decimal digit of the second
number on the list is $a_{22}$, and so forth. As an example, suppose the list of
real numbers between $0$ and $1$ starts out as follows:
$$
0. \ \boxed{2} \ 0 \ 1 \ 4 \ 8 \ 8 \ 0 \ 2 \ \dots \\
0. \ 1 \ \boxed{1} \ 6 \ 6 \ 6 \ 0 \ 2 \ 1 \ \dots \\
0. \ 0 \ 3 \ \boxed{3} \ 5 \ 3 \ 3 \ 2 \ 0 \ \dots \\
0. \ 9 \ 6 \ 7 \ \boxed{7} \ 6 \ 8 \ 0 \ 9 \ \dots \\
0. \ 0 \ 0 \ 0 \ 3 \ \boxed{1} \ 0 \ 0 \ 2 \ \dots
$$
The diagonal elements are boxed: $a_{11}$ is $2$, $a_{22}$ is $1$, $a_{33}$ is
$3$, $a_{44}$ is $7$, $a_{55}$ is $1$, and so forth.
Construct a new decimal number $d = 0.d_1d_2d_3\cdots d_n \cdots$ as follows:
$$
d_n =
\begin{cases}
1 & \text{if } a_{nn} \neq 1 \\
2 & \text{if } a_{nn} = 1
\end{cases}
$$
In the previous example,
$$
d_1 \text{ is } 1 \text{ because } a_{11} = 2 \neq 1,\\
d_2 \text{ is } 2 \text{ because } a_{22} = 1,\\
d_3 \text{ is } 1 \text{ because } a_{33} = 3 \neq 1,\\
d_4 \text{ is } 1 \text{ because } a_{44} = 7 \neq 1,\\
d_5 \text{ is } 2 \text{ because } a_{55} = 1,
$$
and so forth. Hence $d$ would equal $0.12112\dots$.
The crucial observation is that for _each integer $n$, $d$ differs in the $n$th
decimal position from the $n$th number on the list._ But this implies that $d$
is not on the list! In other words, $d$ is a real number between $0$ and $1$
that is not on the list of _all_ real numbers between $0$ and $1$. This
contradiction shows the falseness of the supposition that the set of all numbers
between $0$ and $1$ is countable. Hence the set of all real numbers between $0$
and $1$ is uncountable _[as was to be shown]._
---
Page 503
**Theorem 7.4.3**
Any subset of any countable set is countable.
**Proof:**
Let $A$ be a particular but arbitrarily chosen countable set and let $B$ be any
subset of $A$. _[We must show that $B$ is countable.]_ Either $B$ is finite or
it is infinite. If $B$ is finite, then $B$ is countable by the definition of
countable, and we are done. So suppose $B$ is infinite. Since $A$ is countable,
the distinct elements of $A$ can be represented as a sequence
$$ a_1, a_2, a_3, \dots $$
Define a function $g: \mathbb{Z}^+ \to B$ inductively as follows:
1. Search sequentially through elements of $a_1, a_2, a_3, \dots$ until an
element of $B$ is found _[This must happen eventually since $B \subseteq A$
and $B \neq \emptyset$.]_ Call that element $g(1)$.
2. For each integer $k \geq 2$, suppose $g(k - 1)$ has been defined. Then
$g(k - 1) = a_i$ form some $a_i$ in $\{a_1, a_2, a_3, \dots\}$. Starting with
$a_i + 1$, search sequentially through $a_i + 1, a_i + 2, a_i + 3, \dots$
trying to find an element of $B$. One must be found eventually because $B$ is
infinite, and $\{g(1), g(2), \dots, g(k - 1)\}$ is a finite set. When an
element of $B$ is found, define it to be $g(k)$.
By (1) and (2) above, the function $g$ is defined for each positive integer.
Since the elements of $a_1, a_2, a_3, \dots$ are all distinct, $g$ is
one-to-one. Furthermore, the searches for elements of $B$ are sequential: Each
picks up where the previous one left off. Thus every element of $A$ is reached
during some search. Moreover, all the elements of $B$ are located somewhere in
the sequence $a_1, a_2, a_3, \dots$, and so every element of $B$ is eventually
found and made the image of some integer. Hence $g$ is onto. These remarks show
that $g$ is a one-to-one correspondence from $\mathbb{Z}^+$ to $B$. So $B$ is
countably infinite and thus countable _[as was to be shown]._
---
Page 504
**Corollary 7.4.4**
Any set with an uncountable subset is uncountable.
**Proof:**
Consider the following equivalent phrasing of Theorem 7.4.3: For every set $S$
and for every subset $A$ of $S$, if $S$ is countable, then $A$ is countable. The
contrapositive of this statement is logically equivalent to it and states: For
every set $S$ and for every subset $A$ of $S$, if $A$ is uncountable then $S$ is
uncountable. Since this is an equivalent phrasing for the corollary, the
corollary is proved.

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chapter_7/test_yourself.md
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@ -151,3 +151,38 @@ for some $x_1, x_2 \in X$, $(g \circ f)(x_1) = (g \circ f)(x_2)$; $x_1 = x_2$
then showing that _____.
for some $z \in Z$, there exists some $x \in X$, such that $(g \circ f)(x) = z$
---
Page 507
**Test Yourself**
1. A set is finite if, and only if, _____.
2. To prove that a set $A$ has the same cardinality as a set $B$ you must _____.
3. The reflexive property of cardinality says that given any set $A$, _____.
4. The symmetric property of cardinality says that given any sets $A$ and $B$,
_____.
5. The transitive property of cardinality says that given any sets $A$, $B$, and
$C$, _____.
6. A set is called countably infinite if, and only if, _____.
7. A set is called countable if, and only if, _____.
8. In each of the following, fill in the blank with the word _countable_ or the
word _uncountable_.
a. The set of all integers is _____.
b. The set of all rational numbers is _____.
c. The set of all real numbers between $0$ and $1$ is _____.
d. The set of all real numbers is _____.
9. The Cantor diagonalization process is used to prove that _____.