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