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chapter_5/exercises.md
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@ -13228,20 +13228,115 @@ Omitted.
16. Give a recursive definition for the set of all strings of $0$'s and $1$'s
that have the same number of $0$'s and $1$'s.
Let $S$ be the set of all strings of $0$'s and $1$'s with the same number of
$0$'s and $1$'s. The following is a recursive definition for $S$.
I. Base: The null string $\lambda \in S$.
II. Recursion: If $s \in S$, then
a. $01s \in S$
b. $s01 \in S$
c. $10s \in S$
d. $s10 \in S$
e. $0s1 \in S$
f. $1s0 \8n S$
III. Restriction: There are no elements of $S$ other than those obtained from
the base and recursion for $S$.
17. Give a recursive definition for the set of all strings of $0$'s and $1$'s
for which all the $0$'s precede all the $1$'s.
Let $S$ be the set of all strings of $0$'s and $1$'s where all the $0$'s precede
all the $1$'s. The following is a recursive definition of $S$.
I. Base: The null string $\lambda \in S$.
II. Recursion: If $s \in S$, then
a. $0s \in S$
b. $01s \in S$
c. $00s \in S$
III. Restriction: There are no elements of $S$ other than those obtained from
the base and recursion for $S$.
18. Give a recursive definition for the set of all strings of $a$'s and $b$'s
that contain an odd number of $a$'s.
Let $S$ be the set of all strings of $a$'s and $b$'s such that they contain an
odd number of $a$'s. The following is a recursive definition of $S$.
I. Base: The null string $\lambda \in S$.
II. Recursion: If $s \in S$, then
a. $sb \in S$
b. $bs \in S$
c. $aas \in S$
d. $asa \in S$
e. $saa \in S$
III. Restriction: There are no elements of $S$ other than those obtained from
the base and recursion for $S$.
19. Give a recursive definition for the set of all strings of $a$'s and $b$'s
that contain exactly one $a$.
Let $S$ be the set of all strings of $a$'s and $b$'s such that they exactly one
$a$. The following is a recursive definition of $S$.
I. Base: The null string $\lambda \in S$.
II. Recursion: If $s \in S$, then
a. $bs \in S$
b. $sb \in S$
III. Restriction: There are no elements of $S$ other than those obtained from
the base and recursion for $S$.
20.
a. Let $A$ be any finite set and let $L$ be the length function on the set of
all strings over $A$. Prove that for every character $a$ in $A$, $L(a) = 1$.
**Proof:**
Suppose $a$ is any character in $A$.
By II(b) of the definition of a string:
$$ L(a) = L(\lambda a) $$
By part 2 of the definition of the length of a string:
$$ = L(\lambda) + L(a) $$
By part 1 of the definition of the length of a string:
$$ = 0 + 1 $$
$$ = 1 $$
Therefore for every character $a$ in $A$, $L(a) = 1$. This is what was to be
shown.
Q.E.D.
b. If $A$ is a finite set, define a set $S$ of strings over $A$ as follows:
I. Base: Every character in $A$ is a string in $S$.
@ -13255,8 +13350,146 @@ recursion.
Use structural induction to prove that given any string $s$ in $S$, the length
of $S$, $L(s)$, is an odd integer.
**Proof (by structural induction):**
Let $P(s)$ be the sentence:
The length of $s$ is an odd integer.
_Basis Step:_
Prove the base definition, $P(a)$, that is:
Every character in $A$ is a string in $S$.
The only strings in the base definition of $S$ are the characters in $A$. By
part (a), we know that the length of all strings in $S$ is equal to $1$
($L(s) = 1$).
$1$ is an odd integer since $1 = 2(0) + 1$.
Therefore $P(a)$ is true.
_Inductive Step:_
Let $s$ be any string, and suppose that $P(s)$ is true, that is:
The length of $s$ is an odd integer.
This is the inductive hypothesis.
We must prove the recursion definition. That is we must prove II:
II. Recursion: If $s$ is any string in $S$, then for every character $c$ in $A$,
$csc$ is a string in $S$.
By the inductive hypothesis, we know that the length of $s$ is odd. By the
definition of odd, it follows that:
$$ L(s) = 2k + 1 $$
for some integer $k$.
By part 2 of the definition of the length of a string:
$$ L(csc) = L(c) + L(s) + L(c) $$
$$ = 1 + L(s) + 1 $$
By substitution:
$$ = 1 + (2k + 1) + 1 $$
$$ = 2k + 1 + 1 + 1 $$
$$ = 2k + 2 + 1 $$
$$ = 2(k + 1) + 1 $$
By the sum of integers and by the definition of odd, $L(csc)$ is odd, and
therefore the recursion definition is true. This is what was to be shown.
_Conclusion:_
Since all strings in the set $S$ are only obtained by the base and recursion
definitions for $S$, we conclude that every character in $A$ is a string in $S$.
21. Write a complete proof for Theorem 5.9.4.
**Proof (by structural induction):**
Let $S$ be the set of all strings over a finite set $A$. Given any string $w$ in
$S$, let the property $P(w)$ be the sentence:
For all strings $u$ and $v$ in $S$, $u(vw) = (uv)w$.
_Basis Step:_
Prove $P(\lambda)$, that is:
For all strings $u$ and $v$ in $S$, $u(v\lambda) = (uv)\lambda$.
By the definition of a string:
$$ u(v\lambda) = u(v) = uv $$
and
$$ (uv)\lambda = (uv) = uv $$
Hence $u(v\lambda) = (uv)\lambda$. Therefore $P(\lambda)$ is true.
_Inductive Step:_
Let $w$ be any string, and suppose $P(w)$, that is:
For all strings $u$ and $v$ in $S$, $u(vw) = (uv)w$.
Let $y$ be a string, and suppose that $y$ is obtained from $w$ by applying a
rule from the recursion for $S$.
This is the inductive hypothesis.
We must prove $P(y)$, that is:
For all strings $u$ and $v$ in $S$, $u(vy) = (uv)y$.
By part II(a) of the definition of a string:
$$ wc \in S $$
for some character $c$ in $A$.
By II(c) of the definition of a string:
$$ u(vy) = u(vwc) $$
By part II(c) of the definition of a string:
$$ = u(vw)c $$
By substitution of the inductive hypothesis:
$$ = ((uv)w)c $$
By part II(c) of the definition of a string:
$$ = (uv)(wc) $$
By substitution of the inductive hypothesis:
$$ = (uv)y $$
This is what was to be shown.
_Conclusion:_
Since there is no string in $S$ other than objects obtained from the base and
the recursion, we conclude that if $u$, $v$, and $w$ are strings in $S$, then
$u(vw) = (uv)w$.
Q.E.D.
22. If $S$ is the set of all strings over a finite set $A$ and if $u$ is any
string in $S$, define the _string reversal function_, $\text{Rev}$, as
follows:
@ -13269,34 +13502,121 @@ $\text{Rev}(ua) = a\text{Rev}(u)$.
Use structural induction to prove that for all strings $u$ and $v$ in $S$,
$\text{Rev}(uv) = \text{Rev}(v)\text{Rev}(u)$.
**Proof (by structural induction):**
Let $P(v)$ be the sentence:
$\text{Rev}(uv) = \text{Rev}(v)\text{Rev}(u)$.
_Basis Step:_
Prove $P(\lambda)$, that is:
$\text{Rev}(u\lambda) = \text{Rev}(\lambda)\text{Rev}(u)$.
By part (b) we know that:
$$ = \lambda\text{Rev}(u) $$
By part (a) we know that:
$$ \text{Rev}(\lambda) = \lambda $$
By substitution:
$$ = \text{Rev}(\lambda)\text{Rev}(u) $$
This is what was to be shown, therefore $P(\lambda)$ is true.
_Inductive Step:_
Let $v$ is any string, and suppose $P(v)$, that is:
$\text{Rev}(uv) = \text{Rev}(v)\text{Rev}(u)$.
This is the inductive hypothesis.
Let $y$ be some string obtained by the recursive definition for $v$.
We must prove $P(y)$, that is:
$\text{Rev}(uy) = \text{Rev}(y)\text{Rev}(u)$.
By substitution of the inductive hypothesis and II(a) of the definition of a
string:
$$ \text{Rev}(uy) = \text{Rev}(u(vc)) $$
for some character $c$ in $A$.
By part (b):
$$ = (vc)\text{Rev}(u) $$
By substitution of part (a)
$$ = \text{Rev}(vc)\text{Rev}(u) $$
BY substitution of the inductive hypothesis:
$$ = \text{Rev}(y)\text{Rev}(u) $$
This is what was to be shown. Therefore $P(y)$ is true.
_Conclusion:_
Since all strings in $S$ are obtained by the base and recursion definitions for
$S$, we conclude that for all strings $u$ and $v$ in $S$,
$\text{Rev}(uv) = \text{Rev}(v)\text{Rev}(u)$.
Q.E.D.
23. Use the definition of McCarthy's 91 function in Example 5.9.7 to show the
following:
a. $M(86) = M(91)$
Omitted.
b. $M(91) = 91$
Omitted.
24. Prove that McCarthy's 91 function equals $91$ for all positive integers less
than or equal to $101$.
Omitted.
25. Use the definition of the Ackermann function in Example 5.9.8 to compute the
following:
a. $A(1, 1)$
Omitted.
b. $A(2, 1)$
Omitted.
26. Use the definition of the Ackermann function to show the following:
a. $A(1, n) = n + 2$, for each nonnegative integer $n$
Omitted.
b. $A(2, n) = 3 + 2n$, for each nonnegative integer $n$
Omitted.
c. $A(3, n) = 8 \cdot 2^n - 3$, for each nonnegative integer $n$
Omitted.
27. Compute $T(2), T(3), T(4), T(5), T(6)$, and $T(7)$ for the "function" $T$
defined after Example 5.9.9.
Omitted.
28. Student $A$ tries to define a function: $F: \mathbb{Z}^+ \to \mathbb{Z}$ by
the rule
@ -13312,6 +13632,8 @@ $$
for each integer $n \geq 1$. Student $B$ claims that $F$ is not well defined.
Justify student $B$'s claim.
Omitted.
29. Student $C$ tries to define a function $G: \mathbb{Z}^+ \to \mathbb{Z}$ by
the rule
@ -13326,3 +13648,5 @@ $$
for each integer $n \geq 1$. Student $D$ claims that $G$ is not well defined.
Justify student $D$'s claim.
Omitted.