🚧 Setup for 5.9
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@ -12292,3 +12292,320 @@ c. Express the explicit formula for the Fibonacci sequence in terms of $\phi_1$
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and $\phi_2$.
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Omitted.
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---
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Page 397
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**Exercise Set 5.9**
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1. Consider the set of Boolean expressions defined in Example 5.9.1. Give
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derivations showing that each of the following is a Boolean expression over
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the English alphabet $\{a, b, c, \dots, x, y, z\}$.
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a. $\neg p \vee (q \wedge (r \vee \neg s))$
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b. $(p \vee q) \vee \neg((p \wedge \neg s) \wedge r)$
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2. Consider the set $C$ of parenthesis structures defined in Example 5.9.2. Give
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derivations showing that each of the following is in $C$.
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a. $()(())$
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b. $(())(())$
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3. Let $S$ be the set of all strings over a finite set $A$ and let $a$, $b$, and
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$c$ be any characters in $A$.
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a. Using Theorem 5.9.1 but not Theorem 5.9.3 or 5.9.4, show that
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$(ab)c = a(bc)$.
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b. Show that $ab$ is a string in $S$. Then use the result of part (a) to
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conclude that $a(bc)$ is a string in $S$.
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(This exercise shows that parentheses are not needed when writing the string
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$abc$.)
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4. Consider the _MIU_-system discussed in Example 5.9.4. Give derivations
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showing that each of the following is in the _MIU_-system.
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a. MIUI
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b. MUIIU
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5. The set of arithmetic expressions over the real numbers can be defined
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recursively as follows:
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I. Base: Each real number $r$ is an arithmetic expression.
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II. Recursion: If $u$ and $v$ are arithmetic expressions, then the following are
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also arithmetic expressions:
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(a) $(+u)$
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(b) $(-u)$
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\(c\) $(u + v)$
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(d) $(u - v)$
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(e) $(u \cdot v)$
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(f) $\left(\frac{u}{v}\right)$
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III. Restriction: There are no arithmetic expressions over the real numbers
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other than those obtained from I and II.
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(Note that the _expression $\left(\dfrac{u}{v}\right)$ is allowed to be an
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arithmetic expression even though the value of $v$ may be $0$.) Give the
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derivations showing that each of the following is an arithmetic expression.
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a. $((2 \cdot (0.3 - 4.2)) + (-7))$
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b. $\left(\frac{(9 \cdot(6 \cdot 1 + 2))}{((4 - 7) \cdot 6)}\right)$
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6. Let $S$ be a set of integers defined recursively as follows:
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I. Base: $5$ is in $S$.
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II. Recursion: Given any integer $n$ in $S$, $n + 4$ is in $S$.
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III. Restriction: No integers are in $S$ other than those derived from rules I
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and II above.
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Use structural induction to prove that for every integer $n$ in $S$,
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$n \mod 2 = 1$.
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7. Define a set $S$ of strings over the set $\{0, 1\}$ recursively as follows:
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I. Base: $1 \in S$
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II. Recursion: If $s \in S$, then
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(a) $0s \in S$
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(b) $1s \in S$
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III. Restriction: Nothing is in $S$ other than objects defined in I and II
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above.
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Use structural induction to prove that every string in $S$ ends in a $1$.
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8. Define a set $S$ of strings over the set $\{a, b\}$ recursively as follows:
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I. Base $a \in S$
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II. Recursion: If $s \in S$, then
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(a) $sa \in S$
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(b) $sb \in S$
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III. Restriction: Nothing is in $S$ other than objects defined in I and II
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above.
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Use structural induction to prove that every string in $S$ begins with an $a$.
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9. Define a set $S$ of strings over the set $\{a, b\}$ recursively as follows:
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I. Base: $\lambda \in S$
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II. Recursion: If $s \in S$, then
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(a) $bs \in S$
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(b) $sb \in S$
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\(c\) $saa \in S$
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(d) $aas \in S$
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III. Restriction: Nothing is in $S$ other than objects defined in I and II
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above.
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Use structural induction to prove that every string in $S$ contains an even
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number of $a$'s.
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10. Define a set $S$ of strings over the set of all integers recursively as
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follows:
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I. Base
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$1 \in S, 2 \in S, 3 \in S, 4 \in S, 5 \in S, 6 \in S, 7 \in S, 8 \in S, 9 \in S$
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II. Recursion: If $s \in S$ and $t \in S$, then
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(a) $s0 \in S$
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(b) $st \in S$
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III. Restriction: Nothing is in $S$ other than objects defined in I and II
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above.
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Use structural induction to prove that no string in $S$ represents an integer
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with a leading zero.
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11. Define a set $S$ of strings over the set of all integers recursively as
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follows:
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I. Base: $1 \in S, 3 \in S, 5 \in S, 7 \in S, 9 \in S$
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II. Recursion: If $s \in S$ and $t \in S$, then
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(a) $st \in S$
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(b) $2s \in S$
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\(c\) $4s \in S$
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(d) $6s \in S$
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(e) $8s \in S$
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III. Restriction: Nothing is in $S$ other than objects defined in I and II
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above.
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Use structural induction to prove that every string in $S$ represents an odd
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integer when written in decimal notation.
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12. Define a set $S$ of integers recursively as follows:
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I. Base: $0 \in S, 5 \in S$
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II. Recursion: If $k \in S$ and $p \in S$, then
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(a) $k + p \in S$
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(b) $k - p \in S$
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III. Restriction: Nothing is in $S$ other than objects defined in I and II
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above.
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Use structural induction to prove that every integer in $S$ is divisible by $5$.
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13. Define a set $S$ of integers recursively as follows:
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I. Base: $0 \in S$
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II. Recursion: If $k \in S$, then
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(a) $k + 3 \in S$
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(b) $k - 3 \in S$
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III. Restriction: Nothing is in $S$ other than objects defined in I and II
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above.
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Use structural induction to prove that every integer in $S$ is divisible by $3$.
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14. Is the string _MU_ in the _MIU_-system? Use structural induction to prove
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your answer.
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15. Determine whether either of the following parenthesis configuration is in
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the set $c$ defined in Example 5.9.2. Use structural induction to prove your
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answers.
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a. $()(()$
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b. $(()()))(()$
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16. Give a recursive definition for the set of all strings of $0$'s and $1$'s
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that have the same number of $0$'s and $1$'s.
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17. Give a recursive definition for the set of all strings of $0$'s and $1$'s
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for which all the $0$'s precede all the $1$'s.
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18. Give a recursive definition for the set of all strings of $a$'s and $b$'s
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that contain an odd number of $a$'s.
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19. Give a recursive definition for the set of all strings of $a$'s and $b$'s
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that contain exactly one $a$.
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20.
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a. Let $A$ be any finite set and let $L$ be the length function on the set of
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all strings over $A$. Prove that for every character $a$ in $A$, $L(a) = 1$.
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b. If $A$ is a finite set, define a set $S$ of strings over $A$ as follows:
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I. Base: Every character in $A$ is a string in $S$.
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II. Recursion: If $s$ is any string in $S$, then for every character $c$ in $A$,
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$csc$ is a string in $S$.
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III. Restriction Nothing is in $S$ except strings obtained from the base and the
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recursion.
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Use structural induction to prove that given any string $s$ in $S$, the length
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of $S$, $L(s)$, is an odd integer.
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21. Write a complete proof for Theorem 5.9.4.
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22. If $S$ is the set of all strings over a finite set $A$ and if $u$ is any
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string in $S$, define the _string reversal function_, $\text{Rev}$, as
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follows:
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a. $\text{Rev}(\lambda) = \lambda$
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b. For every string $u$ in $S$ and for every character $a$ in $A$,
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$\text{Rev}(ua) = a\text{Rev}(u)$.
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Use structural induction to prove that for all strings $u$ and $v$ in $S$,
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$\text{Rev}(uv) = \text{Rev}(v)\text{Rev}(u)$.
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23. Use the definition of McCarthy's 91 function in Example 5.9.7 to show the
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following:
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a. $M(86) = M(91)$
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b. $M(91) = 91$
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24. Prove that McCarthy's 91 function equals $91$ for all positive integers less
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than or equal to $101$.
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25. Use the definition of the Ackermann function in Example 5.9.8 to compute the
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following:
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a. $A(1, 1)$
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b. $A(2, 1)$
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26. Use the definition of the Ackermann function to show the following:
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a. $A(1, n) = n + 2$, for each nonnegative integer $n$
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b. $A(2, n) = 3 + 2n$, for each nonnegative integer $n$
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c. $A(3, n) = 8 \cdot 2^n - 3$, for each nonnegative integer $n$
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27. Compute $T(2), T(3), T(4), T(5), T(6)$, and $T(7)$ for the "function" $T$
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defined after Example 5.9.9.
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28. Student $A$ tries to define a function: $F: \mathbb{Z}^+ \to \mathbb{Z}$ by
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the rule
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$$
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F(n) =
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\begin{cases}
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1 & \text{if } n \text{ is } 1 \\
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F\left(\dfrac{n}{2}\right) & \text{if } n \text{ is even} \\
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1 + F(5n - 9) & \text{if } n \text{ is odd and } n > 1
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\end{cases}
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$$
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for each integer $n \geq 1$. Student $B$ claims that $F$ is not well defined.
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Justify student $B$'s claim.
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29. Student $C$ tries to define a function $G: \mathbb{Z}^+ \to \mathbb{Z}$ by
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the rule
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$$
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G(n) =
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\begin{cases}
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1 & \text{if } n \text{ is } 1 \\
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G\left(\dfrac{n}{2}\right) & \text{if } n \text{ is even} \\
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2 + G(3n - 5) & \text{if } n \text{ is odd and } n > 1
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\end{cases}
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$$
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for each integer $n \geq 1$. Student $D$ claims that $G$ is not well defined.
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Justify student $D$'s claim.
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