diff --git a/chapter_7/exercises.md b/chapter_7/exercises.md index ee6f302..f23ce89 100644 --- a/chapter_7/exercises.md +++ b/chapter_7/exercises.md @@ -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.) diff --git a/chapter_7/notes.md b/chapter_7/notes.md index 912f301..9d6f1e3 100644 --- a/chapter_7/notes.md +++ b/chapter_7/notes.md @@ -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. diff --git a/chapter_7/test_yourself.md b/chapter_7/test_yourself.md index 169329e..809019b 100644 --- a/chapter_7/test_yourself.md +++ b/chapter_7/test_yourself.md @@ -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 _____.