From bddf147619a61c4662a4f89150e9632a94928eca Mon Sep 17 00:00:00 2001 From: tomit4 Date: Mon, 22 Jun 2026 19:40:38 -0700 Subject: [PATCH] :construction: Setup for 5.3 --- chapter_5/exercises.md | 323 +++++++++++++++++++++++++++++++++++++ chapter_5/notes.md | 238 +++++++++++++++++++++++++++ chapter_5/test_yourself.md | 10 ++ 3 files changed, 571 insertions(+) diff --git a/chapter_5/exercises.md b/chapter_5/exercises.md index 0517efe..18eae68 100644 --- a/chapter_5/exercises.md +++ b/chapter_5/exercises.md @@ -2820,3 +2820,326 @@ Omitted. consecutive integers is divisible by $p$. Omitted. + +--- + +**Exercise Set 5.3** + +Page 320 + +1. Use mathematical induction (and the proof of proposition 5.3.1 as a model) to + show that any amount of money of at least 14¢ can be made up using 3¢ and 8¢ + coins. + +2. Use mathematical induction to show that any postage of at least 12¢ can be + obtained using 3¢ and 7¢ stamps. + +3. Stamps are sold in packages containing either 5 stamps or 8 stamps. + +a. Show that a person can obtain 5, 8, 10, 13, 15, 16, 20, 21, 24, or 25 stamps +by buying a collection of 5-stamp packages and 8-stamp packages. + +b. Use mathematical induction to show that any quantity of at least 28 stamps +can be obtained by buying a collection of 5-stamp packages and 8-stamp packages. + +4. For each positive integer $n$, let $P(n)$ be the sentence that describes the + following divisibility property: + +$$ 5^n - 1 \text{ is divisible by } 4 $$ + +a. Write $P(0)$. Is $P(0)$ true? + +b. Write $P(k)$. + +c. Write $P(k + 1)$. + +d. In a proof by mathematical induction that this divisibility property holds +for every integer $n \geq 0$, what must be shown in the inductive step? + +5. For each positive integer $n$, let $P(n)$ be the inequality + +$$ 2^n < (n + 1)! $$ + +a. Write $P(2)$. Is $P(2)$ true? + +b. Write $P(k)$. + +c. Write $P(k + 1)$. + +d. In a proof by mathematical induction that this inequality holds for every +integer $n \geq 2$, what must be shown in the inductive step? + +6. For each positive integer $n$, let $P(n)$ be the sentence + +Any checkerboard with dimensions $2 \times 3n$ can be completely covered with +L-shaped trominoes. + +a. Write $P(1)$. Is $P(1)$ true? + +b. Write $P(k)$. + +c. Write $P(k + 1)$. + +d. In a proof by mathematical induction that $P(n)$ is true for each integer +$n \geq 1$, what must be shown in the inductive step? + +7. For each positive integer $n$, let $P(n)$ be the sentence + +In any round-robin tournament involving $n$ teams, the teams can be labeled +$T_1$, $T_2$, $T_3$, \dots, $T_n$, so that $T_i$ beats $T_{i + 1}$ for every +$i = 1, 2, \dots, n$. + +a. Write $P(2)$. Is $P(2)$ true? + +b. Write $P(k)$. + +c. Write $P(k + 1)$. + +d. In a proof by mathematical induction that $P(n)$ is true for each integer +$n \geq 2$, what must be shown in the inductive step? + +Prove each statement in 8-23 by mathematical induction. + +8. $5^n - 1$ is divisible by $4$, for every integer $n \geq 0$. + +9. $7^n - 1$ is divisible by $6$, for every integer $n \geq 0$. + +10. $n^3 - 7n + 3$ is divisible by $3$, for each integer $n \geq 0$. + +11. $3^{2n} - 1$ is divisible by $8$, for every integer $n \geq 0$. + +12. For any integer $n \geq 0$, $7^n - 2^n$ is divisible by $5$. + +13. For any integer $n \geq 0$, $x^n -y^n$ is divisible by $x - y$, where $x$ + and $y$ are any integers with $x \neq y$. + +14. $n^3 - n$ is divisible by $6$, for each integer $n \geq 0$. + +15. $n(n^2 + 5)$ is divisible by $6$, for each integer $n \geq 0$. + +16. $2^n < (n + 1)!$, for every integer $n \geq 2$. + +17. $1 + 3n \leq 4^n$, for every integer $n \geq 0$. + +18. $5^n + 9 < 6^n$, for each integer $n \geq 2$. + +19. $n^2 < 2^n$, for every integer $n \geq 5$. + +20. $2^n < (n + 2)!$, for each integer $n \geq 0$. + +21. $\sqrt{n} < \dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + \dots + \dfrac{1}{\sqrt{n}}$, + for every integer $n \geq 2$. + +22. $1 + nx \leq (1 + x)^n$, for every real number $x > -1$ and every integer + $n \geq 2$. + +23. + +a. $n^3 > 2n + 1$, for each integer $n \geq 2$. + +b. $n! > n^2$, for each integer $n \geq 4$. + +24. A sequence $a_1, a_2, a_3, \dots$ is defined by letting $a_1 = 3$ and + $a_k = 7a_{k - 1}$ for each integer $k \geq 2$. Show that + $a_n = 3 \cdot 7^{n - 1}$ for every integer $n \geq 1$. + +25. A sequence $b_0, b_1, b_2, \dots$ is defined by letting $b_0 = 5$ and + $b_k = 4 + b_{k - 1}$ for each integer $k \geq 1$. Show that $b_n > 4n$ for + every integer $n \geq 0$. + +26. A sequence $c_0, c_1, c_2, \dots$ is defined by letting $c_0 = 3$ and + $c_k = (c_{k - 1})^2$ for every integer $k \geq 1$. Show that $c_n = 3^{2n}$ + for each integer $n \geq 0$. + +27. A sequence $d_1, d_2, d_3, \dots$ is defined by letting $d_1 = 2$ and + $d_k = \dfrac{d_{k - 1}}{k}$ for each integer $k \geq 2$. Show that for + every integer $n \geq 1$, $d_n = \dfrac{2}{n!}$. + +28. Prove that for every integer $n \geq 1$, + +$$ \frac{1}{3} = \frac{1 + 3 + 5 + \dots + (2n - 1)}{(2n + 1)(2n + 3) + \dots + (2n + (2n - 1))} $$ + +Exercises 29 and 30 use the definition of string and string length from page 13 +in Section 1.4. Recursive definitions for these terms are given in section 5.9. + +29. A set $L$ consists of strings obtained by juxtaposing one or more of _abb_, + _bab_, and _bba_. Use mathematical induction to prove that for every integer + $n \geq 1$, if a string $s$ in $L$ has a length $3n$, then $s$ contains an + even number of _b_'s. + +30. A set $S$ consists of strings obtgained by juxtaposing one or more copies of + 1110 and 0111. Use mathematical induction to prove that for every integer + $n \geq 1$, if a string $s$ in $S$ has a length $4n$, then the number of 1's + in $s$ is a multiple of 3. + +31. Use mathematical induction to give an alternative proof for the statement + proved in Example 4.9.9: + +For any positive integer $n$, a complete graph on $n$ vertices has +$\dfrac{n(n - 1)}{2}$ edges. _Hint:_ Let $P(n)$ be the sentence, "the number of +edges in a complete graph on $n$ vertices is $\dfrac{n(n - 1)}{2}$." + +32. Some $5 \times 5$ checkerboards with one square removed can be completely + covered by L-shaped trominoes, whereas other $5 \times 5$ checkerboards + cannot. Find examples of both kinds of checkerboards. Justify your answers. + +33. Consider a $4 \times 6$ checkerboard. Draw a covering of the board by + L-shaped trominoes. + +34. + +a. Use mathematical induction to prove that for each integer $n \geq 1$, any +checkerboard with dimensions $2 \times 3n$ can be completely covered with +L-shaped trominoes. + +b. Let $n$ be any integer greater than or equal to $1$. Use the result of part +(a) to prove by mathematical induction that for every integer $m$, any +checkerboard with dimensions $2m \times 3n$ can be completely covered with +L-shaped trominoes. + +35. Let $m$ and $n$ be any integers that are greater than or equal to $1$. + +a. Prove that a necessary condition for an $m \times n$ checkerboard to be +completely coverable by L-shaped trominoes is that $mn$ be divisible by $3$. + +b. Prove that having $$ be divisible by $3$ is not a sufficient condition for an +$m \times n$ checkerboard to be completely covered by L-shaped trominoes. + +36. In a round-robin tournament each team plays every other team exactly once + with ties not allowed. If the teams are labeled $T_1, T_2, \dots, T_n$, then + the outcome of such a tournament can be represented by a directed graph, in + which the teams are represented as dots and an arrow is drawn from one dot + to another if, and only if, the following team represented by the first dot + beats the team represented by the second dot. For example, the following + directed graph shows one outcome of a round-robin tournament involving five + teams, A, B, C, D, and E. + +See Page 322 for image. + +Use mathematical induction to show that in any round-robin tournament involving +$n$ teams, where $n \geq 2$, it is possible to label the teams +$T_1, T_2, \dots, T_n$ so that $T_i$ beats $T_{i + 1}$ for all +$i = 1, 2, \dots n - 1$,. (For instance, one such labeling in the example above +is $T_1 = 1, T_2 = B, T_3 = C, T_4 = E, T_5 = D$.) (_Hint:_ Given $k + 1$ teams, +pick one - say $T'$ - and apply the inductive hypothesis to the remaining teams +to obtain an ordering $T_1, T_2, \dots, T_k$. Consider three cases: $T'$ beats +$T_1$, $T'$ loses to the first $m$ teams (where $1 \leq m \leq k - 1$) and beats +the $(m + 1)$st team, and $T'$ loses to all the other teams.) + +37. On the outside rim of a circular disk the integers from $1$ through $30$ are + painted in random order. Show that no matter what this order is, there must + be three successive integers whose sum is at least 45. + +38. Suppose that $n$ _a_'s and $n$ _b_'s are distributed around the outside of a + circle. Use mathematical induction to prove that for any integer $n \geq 1$, + given any such arrangement, it is possible to find a starting point so that + if you travel around the circle in a clock-wise direction, the number of + _a_'s you pass is never less than the number of _b_'s you have passed. For + example, in the diagram shown below, you could start at the _a_ with an + asterisk. + +See Page 322 for image. + +39. For a polygon to be **convex** means that given any two points on or inside + the polygon, the line joining the points lies entirely inside the polygon. + Use mathematical induction to prove that for every integer $n \geq 3$, the + angles of any $n$-sided convex polygon add up to $180(n - 2)$ degrees. + +40. + +a. Prove that in an $8 \times 8$ checkerboard with alternating black and white +squares, if the squares in the top right and bottom left corners are removed the +remaining board cannot be covered with dominoes. (_Hint:_ Mathematical induction +is not needed for this proof.) + +b. Use mathematical induction to prove that for each positive integer $n$, if a +$2n \times 2n$ checkerboard with alternating black and white squares has one +white square and one black square removed anywhere on the board, the remaining +squares can be covered with dominoes. + +41. A group of people are positioned so that the distance between any two people + is different from the distance between any other two people. Suppose that + the group contains an odd number of people and each person sends a message + to their nearest neighbor. Use mathematical induction to prove that at least + one person does not receive a message from anyone. [This exercise is + inspired by the article "Odd Pie Fights" by L. Carmony, _The Mathematics + Teacher_, **72**(1), 1979, 61-64.] + +42. Show that for any integer $n$, it is possible to find a group of $n$ people + who are all positioned so that the distance between any two people is + different from the distance between any other two people, so that each + person sends a message to their nearest neighbor, and so that every person + in the group receives a message from another person in the group. + +43. Define a game as follows: You begin with an urn that contains a mixture of + white and black balls, and during the game you have access to as many + additional white and black balls as you might need. In each move you remove + two balls from the urn without looking at their colors. If the balls are the + same color, you put in one black ball. If the balls are different colors, + you put the white ball back into the urn and keep the black ball out. + Because each move reduces the number of balls in the urn by one, the game + will end with a single ball in the urn. If you know how many white balls and + how many black balls are initially in the urn, can you predict the color of + the ball at the end of the game? [This exercise is based on one described in + "Why correctness must be a mathematical concern" by E.W. Djikstra, + www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD720.html.] + +a. Map out all possibilities for playing the game starting with two balls in the +urn, then three balls, and then four balls. For each case keep track of the +number of white and black balls you start with and the color of the ball at the +end of the game. + +b. Does the number of white balls seem to be predictive? Does the number of +black balls seem to be predictive? Make a conjecture about the color of the ball +at the end of the game given the numbers of white and black balls at the +beginning. + +c. Use mathematical induction to prove the conjecture you made in part (b). + +44. Let $P(n)$ be the following sentence: Given any graph $G$ with $n$ vertices + satisfying the condition that every vertex of $G$ has degree at most $M$, + then the vertices of $G$ can be colored with at most $M + 1$ colors in such + a way that no two adjacent vertices have the same color. Use mathematical + induction to prove this statement is true for every integer $n \geq 1$. + +In order for a proof by mathematical induction to be valid, the basis statement +must be true for $n = a$ and the argument of the inductive step must be correct +for every integer $k \geq a$. IN 45 and 46 find the mistakes in the "proofs" by +mathematical induction. + +45. + +**"Theorem:"** For any integer $n \geq 1$, all the numbers in a set of $n$ +numbers are equal to each other. + +**"Proof (by mathematical induction):** It is obviously true that all the +numbers in a set consisting of just one number are equal to each other, so the +basis step is true. For the inductive step, let +$A = \{a_1, a_2, \dots, a_k, a_{k + 1}\}$ be any set of $k + 1$ numbers. Form +two subsets each of size $k$: + +$$ B = \{a_1, a_2, a_3, \dots, a_k\} \text{ and } $$ + +$$ C = \{a_1, a_3, a_4, \dots, a_{k + 1}} $$ + +($B$ consists of all the numbers in $A$ except $a_{k + 1}$, and $C$ consists of +all the numbers in $A$ except $a_2$.) By inductive hypothesis, all the numbers +in $B$ equal $a_1$ and all the numbers in $C$ equal $a_1$ (since both sets have +only $k$ numbers). But every number in $A$ is in $B$ or $C$, so all the numbers +in $A$ equal $a_1$; hence all are equal to each other." + +46. + +**"Theorem:"** For every integer $n \geq 1$, $3^n - 2$ is even. + +**"Proof (by mathematical induction):** Suppose the theorem is true for an +integer $k$, where $k \geq 1$. That is, suppose that $3^k - 2$ is even. We must +show that $3^{k + 1} - 2$ is even. Observe that + +$$ 3^{k + 1} - 2 = 3^k \cdot 3 - 2 = 3^k(1 + 2) - 2 $$ + +$$ = (3^k - 2) + 3^k \cdot 2 $$ + +Now $3^k - 2$ is even by inductive hypothesis and $3^k \cdot 2$ is even by +inspection. Hence the sum of the two quantities is even (by Theorem 4.1.1). It +follows that $3^{k + 1} - 2$ is even, which is what we needed to show." diff --git a/chapter_5/notes.md b/chapter_5/notes.md index c5e07a1..a6c69d4 100644 --- a/chapter_5/notes.md +++ b/chapter_5/notes.md @@ -318,3 +318,241 @@ which is the right-hand side of $P(k + 1)$ _[as was to be shown]._ _[Since we have proved the basis step and the inductive step, we conclude that the theorem is true.]_ + +--- + +Page 314 + +**Proposition 5.3.1** + +For every integer $n \geq 8$, $n$¢ can be obtained using $3$¢ and $5$¢ coins. + +**Proof (by mathematical induction):** + +Let the property $P(n)$ be the sentence + +$n$¢ can be obtained using $3$¢ and $5$¢ coins. + +Show that $P(8)$ is true: + +$P(8)$ is true because $8$¢ can be obtained using one $3$¢ coin and one $5$¢ +coin. + +_[Suppose that $P(k)$ is true for a particular but arbitrarily chosen integer +$k \geq 8$. That is:]_ + +Suppose that $k$ is any integer with $k \geq 8$ such that + +$k$¢ can be obtained using $3$¢ and $5$¢ coins. + +_[We must show that $P(k + 1)$ is true. That is:]_ We must show that + +$(k + 1)$¢ can be obtained using $3$¢ and $5$¢ coins. + +_Case 1 (There is a $5$¢ coin among those that used to make up the $k$¢.):_ + +In this case replace the $5$¢ coin by two $3$¢ coins; the result will be +$(k + 1)$¢. + +_Case 2 (There is not a $5$¢ coin among those used to make up the $k$¢.):_ + +In this case, because $k \geq 8$, at least three $3$¢ coins must have been used. +So remove three $3$¢ coins and replace them by two $5$¢ coins; the result will +be $(k + 1)$¢. + +Thus in either case $(k + 1)$¢ can be obtained using $3$¢ and $5$¢ coins _[as +was to be shown]._ + +_[Since we have proved the basis step and the inductive step, we conclude that +the proposition is true.]_ + +--- + +Page 315 + +**Proposition 5.3.2** + +For each integer $n \geq 0$, $2^{2n} - 1$ is divisible by $3$. + +**Proof (by mathematical induction):** + +Let the property $P(n)$ be the sentence "$2^{2n} - 1$ is divisible by $3$." + +$$ 2^{2n} - 1 \text{ is divisible by } 3 $$ + +_Show that $P(0)$ is true:_ + +To establish $P(0)$, we must show that + +$$ 2^{2 \cdot 0} - 1 \text{ is divisible by 3.} $$ + +But + +$$ 2^{2 \cdot 0} - 1 = 2^0 - 1 = 1 - 1 = 0 $$ + +and $0$ is divisible by $3$ because $0$ = 3 \cdot 0. Hence $P(0)$ is true. + +_Show that for any integer $k \geq 0$, if $P(k)$ is true then $P(k + 1)$ is also +true:_ + +_[Suppose that $P(k)$ is true for a particular but arbitrarily chosen integer +$k \geq 0$. That is:]_ + +Let $k$ be any integer with $k \geq 0$, and suppose that + +$$ 2^{2k} - 1 \text{ is divisible by } 3 $$ + +By definition of divisibility, this means that + +$$ 2^{2k} - 1 = 3r \text{ for some integer } r $$ + +_[We must show that $P(k + 1)$ is true. That is:]_ We must show that + +$$ 2^{2(k + 1)} - 1 \text{ is divisible by } 3 $$ + +Now + +$$ 2^{2(k + 1)} - 1 = 2^{2k + 2} - 1 $$ + +$$ = 2^{2k} \cdot 2^2 - 1 $$ + +$$ = 2^{2k} \cdot 4 - 1 $$ + +$$ = 2^{2k}(3 + 1) - 1 $$ + +$$ = 2^{2k} \cdot 3 + (2^{2k} - 1) $$ + +$$ = 2^{2=} \cdot 3 + 3r $$ + +$$ = 3(2^{2k} + r) $$ + +But $2^{2k} + r$ is an integer because it is a sum of products of integers, and +so, by definition of divisibility, $2^{2(k + 1)} - 1$ is divisible by $3$ _[as +was to be shown]_. + +_[Since we have proved the basis step and the inductive step, we conclude that +the proposition is true.]_ + +--- + +Page 317 + +**Proposition 5.3.3** + +For every integer $n \geq 3$, $2n + 1 < 2^n$. + +**Proof (by mathematical induction):** + +Let the property $P(n)$ be the inequality + +$$ 2n + 1 < 2^n $$ + +_Show that $P(3)$ is true:_ + +To establish $P(3)$, we must show that + +$$ 2 \cdot 3 + 1 < 2^3 $$ + +Now + +$$ 2 \cdot 3 + 1 = 7 \quad \text{ and } \quad 2^3 = 8 \quad \text{ and } \quad 7 < 8 $$ + +Hence $P(3)$ is true. + +_Show that for every integer $k \geq 3$, if $P(k)$ is true then $P(k + 1)$ is +also true:_ + +_[Suppose that $P(k)$ is true for a particular but arbitrarily chosen integer +$k \geq 3$. That is:]_ + +Suppose that $k$ is any integer with $k \geq 3$ such that + +$$ 2k + 1 < 2^k $$ + +_[We must show that $P(k + 1)$ is true. That is:]_ + +We must show that + +$$ 2(k + 1) + 1 < 2^{(k + 1)} $$ + +Now + +$$ 2(k + 1) + 1 = 2k + 1 + 2 $$ + +$$ < 2^k + 2 $$ + +$$ < 2^k + 2^k $$ + +$$ = 2 \cdot 2^k $$ + +$$ = 2^{k + 1} $$ + +Thus by transitivity of order $2(k + 1) + 1 < 2^{k + 1}$ _[as was to be shown]_. + +_[Since we have proved the basis step and the inductive step, we conclude that +the proposition is true.]_ + +--- + +Page 319 + +**Theorem 5.3.4 Covering a Board with Trominoes** + +For any integer $n \geq 1$, if one square is removed from a $2^n \times 2^n$ +checkerboard, the remaining squares can be completely covered by L-shaped +trominoes. + +The main insight leading to a proof of this theorem is the observation that +because $2^{k + 1} = 2 \cdot 2^k$, when a $2^{k + 1} \times 2^{k + 1}$ board is +split in half both vertically and horizontally, each half side will have length +$2^k$ and so each resulting quadrant will be a $2^k \times 2^k$ checkerboard. + +**Proof (by mathematical induction):** + +Let the property $P(n)$ be the sentence + +If any square is removed from a $2^n \times 2^n$ checkerboard, then the +remaining squares can be completely covered by L-shaped trominoes. + +_Show that $P(1)$ is true:_ + +A $2^1 \times 2^1$ checkerboard just consists of four squares. If one square is +removed, the remaining squares form an L, which can be covered by a single +L-shaped tromino, as illustrated in the figure to the left. Hence $P(1)$ is +true. + +_Show that for every integer $k \geq 1$, if $P(k)$ is true then $P(k + 1)$ is +also true:_ + +_[Suppose that $P(k)$ is true for a particular but arbitrarily chosen integer +$k \geq 3$. That is:]_ + +Let $k$ be any integer such that $k \geq 1$, and suppose that + +If any square is removed from a $2^k \times 2^k$ checkerboard, then the +remaining squares can be completely covered by L-shaped trominoes. + +$P(k)$ is the inductive hypothesis. + +_[We must show that $P(k + 1)$ is true. That is:]_ + +We must show that + +If any square is removed from a $2^{k + 1} \times 2^{k + 1}$ checkerboard, then +the remaining squares can be completely covered by L-shaped trominoes. + +Consider a $2^{2k + 1} \times 2^{k + 1}$ checkerboard with one square removed. +Divide it into four equal quadrants: Each will consist of a $2^k \times 2^k$ +checkerboard. In one of the quadrants, one square will have been removed, and +so, by inductive hypothesis, all the remaining squares in this quadrant can be +completely covered by L-shaped trominoes. + +The other three quadrants meet at the center of the checkerboard, and the center +of the checkerboard serves as a corner of a square from each of those quadrants. +An L-shaped tromino can, therefore, be placed on those three central squares. +This situation is illustrated in the figure to the left (see page 320). + +By inductive hypothesis, the remaining squares in each of the three quadrants +can be completely covered by L-shaped trominoes. Thus every square in the +$2^{k + 1} \times 2^{k + 1}$ checkerboard except the one that was removed can be +completely covered by L-shaped trominoes _[as was to be shown]_. diff --git a/chapter_5/test_yourself.md b/chapter_5/test_yourself.md index 8914245..6012e4d 100644 --- a/chapter_5/test_yourself.md +++ b/chapter_5/test_yourself.md @@ -55,3 +55,13 @@ arbitrarily chosen value of an integer $k \geq a$. This supposition is called the _____. One then has to show that _____. $P(k)$ is true; inductive hypothesis; $P(k + 1)$ is true. + +--- + +Page 320 + +1. Mathematical induction differs from the kind of induction used in the natural + sciences because it is actually a form of _____ reasoning. + +2. Mathematical induction can be used to _____ conjectures that have been made + using inductive reasoning.