🚧 Setup for 5.9
This commit is contained in:
tomit4
parent
cd7142bb31
commit
37b6800cd3
3 changed files with 585 additions and 0 deletions
|
|
@ -12292,3 +12292,320 @@ c. Express the explicit formula for the Fibonacci sequence in terms of $\phi_1$
|
|||
and $\phi_2$.
|
||||
|
||||
Omitted.
|
||||
|
||||
---
|
||||
|
||||
Page 397
|
||||
|
||||
**Exercise Set 5.9**
|
||||
|
||||
1. Consider the set of Boolean expressions defined in Example 5.9.1. Give
|
||||
derivations showing that each of the following is a Boolean expression over
|
||||
the English alphabet $\{a, b, c, \dots, x, y, z\}$.
|
||||
|
||||
a. $\neg p \vee (q \wedge (r \vee \neg s))$
|
||||
|
||||
b. $(p \vee q) \vee \neg((p \wedge \neg s) \wedge r)$
|
||||
|
||||
2. Consider the set $C$ of parenthesis structures defined in Example 5.9.2. Give
|
||||
derivations showing that each of the following is in $C$.
|
||||
|
||||
a. $()(())$
|
||||
|
||||
b. $(())(())$
|
||||
|
||||
3. Let $S$ be the set of all strings over a finite set $A$ and let $a$, $b$, and
|
||||
$c$ be any characters in $A$.
|
||||
|
||||
a. Using Theorem 5.9.1 but not Theorem 5.9.3 or 5.9.4, show that
|
||||
$(ab)c = a(bc)$.
|
||||
|
||||
b. Show that $ab$ is a string in $S$. Then use the result of part (a) to
|
||||
conclude that $a(bc)$ is a string in $S$.
|
||||
|
||||
(This exercise shows that parentheses are not needed when writing the string
|
||||
$abc$.)
|
||||
|
||||
4. Consider the _MIU_-system discussed in Example 5.9.4. Give derivations
|
||||
showing that each of the following is in the _MIU_-system.
|
||||
|
||||
a. MIUI
|
||||
|
||||
b. MUIIU
|
||||
|
||||
5. The set of arithmetic expressions over the real numbers can be defined
|
||||
recursively as follows:
|
||||
|
||||
I. Base: Each real number $r$ is an arithmetic expression.
|
||||
|
||||
II. Recursion: If $u$ and $v$ are arithmetic expressions, then the following are
|
||||
also arithmetic expressions:
|
||||
|
||||
(a) $(+u)$
|
||||
|
||||
(b) $(-u)$
|
||||
|
||||
\(c\) $(u + v)$
|
||||
|
||||
(d) $(u - v)$
|
||||
|
||||
(e) $(u \cdot v)$
|
||||
|
||||
(f) $\left(\frac{u}{v}\right)$
|
||||
|
||||
III. Restriction: There are no arithmetic expressions over the real numbers
|
||||
other than those obtained from I and II.
|
||||
|
||||
(Note that the _expression $\left(\dfrac{u}{v}\right)$ is allowed to be an
|
||||
arithmetic expression even though the value of $v$ may be $0$.) Give the
|
||||
derivations showing that each of the following is an arithmetic expression.
|
||||
|
||||
a. $((2 \cdot (0.3 - 4.2)) + (-7))$
|
||||
|
||||
b. $\left(\frac{(9 \cdot(6 \cdot 1 + 2))}{((4 - 7) \cdot 6)}\right)$
|
||||
|
||||
6. Let $S$ be a set of integers defined recursively as follows:
|
||||
|
||||
I. Base: $5$ is in $S$.
|
||||
|
||||
II. Recursion: Given any integer $n$ in $S$, $n + 4$ is in $S$.
|
||||
|
||||
III. Restriction: No integers are in $S$ other than those derived from rules I
|
||||
and II above.
|
||||
|
||||
Use structural induction to prove that for every integer $n$ in $S$,
|
||||
$n \mod 2 = 1$.
|
||||
|
||||
7. Define a set $S$ of strings over the set $\{0, 1\}$ recursively as follows:
|
||||
|
||||
I. Base: $1 \in S$
|
||||
|
||||
II. Recursion: If $s \in S$, then
|
||||
|
||||
(a) $0s \in S$
|
||||
|
||||
(b) $1s \in S$
|
||||
|
||||
III. Restriction: Nothing is in $S$ other than objects defined in I and II
|
||||
above.
|
||||
|
||||
Use structural induction to prove that every string in $S$ ends in a $1$.
|
||||
|
||||
8. Define a set $S$ of strings over the set $\{a, b\}$ recursively as follows:
|
||||
|
||||
I. Base $a \in S$
|
||||
|
||||
II. Recursion: If $s \in S$, then
|
||||
|
||||
(a) $sa \in S$
|
||||
|
||||
(b) $sb \in S$
|
||||
|
||||
III. Restriction: Nothing is in $S$ other than objects defined in I and II
|
||||
above.
|
||||
|
||||
Use structural induction to prove that every string in $S$ begins with an $a$.
|
||||
|
||||
9. Define a set $S$ of strings over the set $\{a, b\}$ recursively as follows:
|
||||
|
||||
I. Base: $\lambda \in S$
|
||||
|
||||
II. Recursion: If $s \in S$, then
|
||||
|
||||
(a) $bs \in S$
|
||||
|
||||
(b) $sb \in S$
|
||||
|
||||
\(c\) $saa \in S$
|
||||
|
||||
(d) $aas \in S$
|
||||
|
||||
III. Restriction: Nothing is in $S$ other than objects defined in I and II
|
||||
above.
|
||||
|
||||
Use structural induction to prove that every string in $S$ contains an even
|
||||
number of $a$'s.
|
||||
|
||||
10. Define a set $S$ of strings over the set of all integers recursively as
|
||||
follows:
|
||||
|
||||
I. Base
|
||||
$1 \in S, 2 \in S, 3 \in S, 4 \in S, 5 \in S, 6 \in S, 7 \in S, 8 \in S, 9 \in S$
|
||||
|
||||
II. Recursion: If $s \in S$ and $t \in S$, then
|
||||
|
||||
(a) $s0 \in S$
|
||||
|
||||
(b) $st \in S$
|
||||
|
||||
III. Restriction: Nothing is in $S$ other than objects defined in I and II
|
||||
above.
|
||||
|
||||
Use structural induction to prove that no string in $S$ represents an integer
|
||||
with a leading zero.
|
||||
|
||||
11. Define a set $S$ of strings over the set of all integers recursively as
|
||||
follows:
|
||||
|
||||
I. Base: $1 \in S, 3 \in S, 5 \in S, 7 \in S, 9 \in S$
|
||||
|
||||
II. Recursion: If $s \in S$ and $t \in S$, then
|
||||
|
||||
(a) $st \in S$
|
||||
|
||||
(b) $2s \in S$
|
||||
|
||||
\(c\) $4s \in S$
|
||||
|
||||
(d) $6s \in S$
|
||||
|
||||
(e) $8s \in S$
|
||||
|
||||
III. Restriction: Nothing is in $S$ other than objects defined in I and II
|
||||
above.
|
||||
|
||||
Use structural induction to prove that every string in $S$ represents an odd
|
||||
integer when written in decimal notation.
|
||||
|
||||
12. Define a set $S$ of integers recursively as follows:
|
||||
|
||||
I. Base: $0 \in S, 5 \in S$
|
||||
|
||||
II. Recursion: If $k \in S$ and $p \in S$, then
|
||||
|
||||
(a) $k + p \in S$
|
||||
|
||||
(b) $k - p \in S$
|
||||
|
||||
III. Restriction: Nothing is in $S$ other than objects defined in I and II
|
||||
above.
|
||||
|
||||
Use structural induction to prove that every integer in $S$ is divisible by $5$.
|
||||
|
||||
13. Define a set $S$ of integers recursively as follows:
|
||||
|
||||
I. Base: $0 \in S$
|
||||
|
||||
II. Recursion: If $k \in S$, then
|
||||
|
||||
(a) $k + 3 \in S$
|
||||
|
||||
(b) $k - 3 \in S$
|
||||
|
||||
III. Restriction: Nothing is in $S$ other than objects defined in I and II
|
||||
above.
|
||||
|
||||
Use structural induction to prove that every integer in $S$ is divisible by $3$.
|
||||
|
||||
14. Is the string _MU_ in the _MIU_-system? Use structural induction to prove
|
||||
your answer.
|
||||
|
||||
15. Determine whether either of the following parenthesis configuration is in
|
||||
the set $c$ defined in Example 5.9.2. Use structural induction to prove your
|
||||
answers.
|
||||
|
||||
a. $()(()$
|
||||
|
||||
b. $(()()))(()$
|
||||
|
||||
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.
|
||||
|
||||
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.
|
||||
|
||||
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.
|
||||
|
||||
19. Give a recursive definition for the set of all strings of $a$'s and $b$'s
|
||||
that contain exactly one $a$.
|
||||
|
||||
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 $A$, $L(a) = 1$.
|
||||
|
||||
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 $S$.
|
||||
|
||||
II. Recursion: If $s$ is any string in $S$, then for every character $c$ in $A$,
|
||||
$csc$ is a string in $S$.
|
||||
|
||||
III. Restriction Nothing is in $S$ except strings obtained from the base and the
|
||||
recursion.
|
||||
|
||||
Use structural induction to prove that given any string $s$ in $S$, the length
|
||||
of $S$, $L(s)$, is an odd integer.
|
||||
|
||||
21. Write a complete proof for Theorem 5.9.4.
|
||||
|
||||
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:
|
||||
|
||||
a. $\text{Rev}(\lambda) = \lambda$
|
||||
|
||||
b. For every string $u$ in $S$ and for every character $a$ in $A$,
|
||||
$\text{Rev}(ua) = a\text{Rev}(u)$.
|
||||
|
||||
Use structural induction to prove that for all strings $u$ and $v$ in $S$,
|
||||
$\text{Rev}(uv) = \text{Rev}(v)\text{Rev}(u)$.
|
||||
|
||||
23. Use the definition of McCarthy's 91 function in Example 5.9.7 to show the
|
||||
following:
|
||||
|
||||
a. $M(86) = M(91)$
|
||||
|
||||
b. $M(91) = 91$
|
||||
|
||||
24. Prove that McCarthy's 91 function equals $91$ for all positive integers less
|
||||
than or equal to $101$.
|
||||
|
||||
25. Use the definition of the Ackermann function in Example 5.9.8 to compute the
|
||||
following:
|
||||
|
||||
a. $A(1, 1)$
|
||||
|
||||
b. $A(2, 1)$
|
||||
|
||||
26. Use the definition of the Ackermann function to show the following:
|
||||
|
||||
a. $A(1, n) = n + 2$, for each nonnegative integer $n$
|
||||
|
||||
b. $A(2, n) = 3 + 2n$, for each nonnegative integer $n$
|
||||
|
||||
c. $A(3, n) = 8 \cdot 2^n - 3$, for each nonnegative integer $n$
|
||||
|
||||
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.
|
||||
|
||||
28. Student $A$ tries to define a function: $F: \mathbb{Z}^+ \to \mathbb{Z}$ by
|
||||
the rule
|
||||
|
||||
$$
|
||||
F(n) =
|
||||
\begin{cases}
|
||||
1 & \text{if } n \text{ is } 1 \\
|
||||
F\left(\dfrac{n}{2}\right) & \text{if } n \text{ is even} \\
|
||||
1 + F(5n - 9) & \text{if } n \text{ is odd and } n > 1
|
||||
\end{cases}
|
||||
$$
|
||||
|
||||
for each integer $n \geq 1$. Student $B$ claims that $F$ is not well defined.
|
||||
Justify student $B$'s claim.
|
||||
|
||||
29. Student $C$ tries to define a function $G: \mathbb{Z}^+ \to \mathbb{Z}$ by
|
||||
the rule
|
||||
|
||||
$$
|
||||
G(n) =
|
||||
\begin{cases}
|
||||
1 & \text{if } n \text{ is } 1 \\
|
||||
G\left(\dfrac{n}{2}\right) & \text{if } n \text{ is even} \\
|
||||
2 + G(3n - 5) & \text{if } n \text{ is odd and } n > 1
|
||||
\end{cases}
|
||||
$$
|
||||
|
||||
for each integer $n \geq 1$. Student $D$ claims that $G$ is not well defined.
|
||||
Justify student $D$'s claim.
|
||||
|
|
|
|||
|
|
@ -1244,3 +1244,249 @@ $$ a_n = Cr^n + Dnr^n $$
|
|||
|
||||
where $C$ and $D$ are the real numbers whose values are determined by the values
|
||||
of $a_0$ and any other known value of the sequence.
|
||||
|
||||
---
|
||||
|
||||
Page 389
|
||||
|
||||
**Recursive Definition for the Set of All Strings over a Finite Set**
|
||||
|
||||
Let $A$ be any finite set. Call the elements of $A$ characters, and define the
|
||||
set $S$ **of all strings over** $A$ as follows:
|
||||
|
||||
I. Base: $\lambda$ is a string in $S$, where $\lambda$ denotes the **null
|
||||
string**, the "string" with no characters.
|
||||
|
||||
II. Recursion: New strings are formed according to the following rules:
|
||||
|
||||
(a) If $u$ is any string in $S$ and if $c$ is any character in $A$, then
|
||||
|
||||
$$ uc \text{ is a string in } S $$
|
||||
|
||||
where $uc$ is called the **concatenation of $u$ and $c$**, and is obtained by
|
||||
appending $c$ on the right of $u$.
|
||||
|
||||
(b) If $u$ is a string in $S$, then both the concatenation of $\lambda$ and $u$,
|
||||
denoted $\lambda u$, and the concatenation of $u$ and $\lambda$, denoted
|
||||
$u\lambda$, are defined to equal $u$. Symbolically:
|
||||
|
||||
$$ \lambda u = u\lambda = u $$
|
||||
|
||||
\(c\) If $u$ and $v$ are any strings in $S$, and if $c$ is any character in $A$,
|
||||
then the concatenation of $u$ and $vc$ is defined to equal the concatenation of
|
||||
$uv$ and $c$. Symbolically:
|
||||
|
||||
$$ u(vc) = (uv)c $$
|
||||
|
||||
III. Restriction: Nothing is a string in $S$ other than objects obtained from
|
||||
the base and the recursion.
|
||||
|
||||
---
|
||||
|
||||
Page 389
|
||||
|
||||
**Theorem 5.9.1 Characters are Strings**
|
||||
|
||||
If $A$ is a finite set and $S$ is the set of all strings over $A$, then every
|
||||
character in $A$ is a string in $S$.
|
||||
|
||||
**Proof:**
|
||||
|
||||
(1) Suppose $c$ is any character in $A$.
|
||||
|
||||
(2) By part I of the definition of string, $\lambda$ is a string in $S$.
|
||||
|
||||
(3) By part II(a) of the definition of string, $\lambda c$ is a string in $S$.
|
||||
|
||||
(4) By part I of the definition of string, $\lambda c = c$.
|
||||
|
||||
(5) Thus $c$ is a string in $S$.
|
||||
|
||||
---
|
||||
|
||||
Page 391
|
||||
|
||||
**Structural Induction for a Recursively Defined Set**
|
||||
|
||||
Let $S$ be a set that has been defined recursively, and let $P(x)$ be a property
|
||||
that objects in $S$ may or may not satisfy. To prove that every object in $S$
|
||||
satisfies $P(x)$, perform the following two steps:
|
||||
|
||||
**Step 1 (basis step):**
|
||||
|
||||
Show that $P(a)$ is true for each object $a$ in the base for $S$.
|
||||
|
||||
**Step 2 (inductive step):**
|
||||
|
||||
Show that for each $x$ in $S$, if $P(x)$ is true and if $y$ is obtained from $x$
|
||||
by applying a rule from the recursion, then $P(y)$ is true. To perform this
|
||||
step,
|
||||
|
||||
**suppose** that $x$ is an arbitrarily chosen element of $S$ for which $P(x)$ is
|
||||
true.
|
||||
|
||||
_[This supposition is the **inductive hypothesis**.]_
|
||||
|
||||
Then
|
||||
|
||||
**show** that if $y$ is obtained from $x$ by applying a rule from the recursion
|
||||
for $S$, then $P(y)$ is true.
|
||||
|
||||
**Conclusion:** Because no objects other than those obtained from the base and
|
||||
recursion are contained in $S$, steps 1 and 2 prove that $P(x)$ is true for
|
||||
every object $x$ in $S$.
|
||||
|
||||
---
|
||||
|
||||
Page 392
|
||||
|
||||
**Definition Length of a String**
|
||||
|
||||
Given the set of all strings $S$ over a finite set $A$, the **length $L$ of a
|
||||
string in $S$** is defined as follows:
|
||||
|
||||
1. $L(\lambda) = 0$.
|
||||
|
||||
2. For every string $u$ in $S$ and for every character $a$ in $A$, the length of
|
||||
$ua$ is one more than the length of $u$. Symbolically:
|
||||
|
||||
$$ L(ua) = L(u) + 1 \quad \text{ where } \quad u \in S \text{ and } a \in A $$
|
||||
|
||||
---
|
||||
|
||||
Page 393
|
||||
|
||||
**Theorem 5.9.2 Additive Property of String Length**
|
||||
|
||||
If $S$ is the set of all strings over a finite set $A$, then for all strings $u$
|
||||
and $v$ in $S$, $L(uv) = L(u) + L(v)$.
|
||||
|
||||
**Proof (by structural induction):**
|
||||
|
||||
Let $S$ be the set of all strings over a finite set $A$. Given any string $v$ in
|
||||
$S$, let the property $P(v)$ be the sentence
|
||||
|
||||
For every string $u$ in $S$, $L(uv) = L(u) + L(v)$.
|
||||
|
||||
We will show that $P(v)$ is true for every string $v$ in $S$.
|
||||
|
||||
_Show that $P(a)$ is true for each string $a$ in the base for $S$:_
|
||||
|
||||
The only string in the base for $S$ is $\lambda$, and if $u$ is any string in
|
||||
$S$, then
|
||||
|
||||
$$ L(u\lambda) = L(u) $$
|
||||
|
||||
$$ = L(u) + 0 $$
|
||||
|
||||
$$ = L(u) + L(\lambda) $$
|
||||
|
||||
This shows that $P(\lambda)$ is true.
|
||||
|
||||
_Show that for each string $x$ in $S$, if $P(x)$ is true and if $y$ is obtained
|
||||
from $x$ by applying a rule from the recursion for $S$, then $P(y)$ is true:_
|
||||
|
||||
The recursion for $S$ consists of three rules denoted II(a), II(b), II\(c\), but
|
||||
rule II(a) is the only one that generates new strings in $S$. Suppose $v$ is any
|
||||
string in $S$ such that $P(v)$ is true. In other words, suppose that
|
||||
$L(uv) = L(u) + L(v)$. _[This is the inductive hypothesis.]_
|
||||
|
||||
When rule II(a) is applied to $v$, the result is $vc$, where $c$ is a character
|
||||
in $A$. So, to complete the inductive step, we must show that $P(vc)$ is true.
|
||||
Now
|
||||
|
||||
$$ L(u(vc)) = L((uv)c) $$
|
||||
|
||||
$$ = L(uv) + 1 $$
|
||||
|
||||
$$ = (L(u) + L(v)) + 1 $$
|
||||
|
||||
$$ = L(u) + (L(v) + 1) $$
|
||||
|
||||
$$ = L(u) + L(vc) $$
|
||||
|
||||
Hence $P(vc)$ is true _[as was to be shown]_.
|
||||
|
||||
_Conclusion:_
|
||||
|
||||
Because there are no strings in $S$ other than those obtained through the base
|
||||
and the recursion for $S$, we conclude that every string in $S$ satisfies the
|
||||
additive property for string length.
|
||||
|
||||
---
|
||||
|
||||
Page 394
|
||||
|
||||
**Theorem 5.9.3 The Concatenation of Any Two Strings is a String**
|
||||
|
||||
If $S$ is the set of all strings over a finite set $A$ and $u$ and $v$ are any
|
||||
strings in $S$, then $uv$ is a string in $S$.
|
||||
|
||||
**Proof (by structural induction):**
|
||||
|
||||
Let $S$ be the set of all strings over a finite set $A$. Given any string $v$ in
|
||||
$S$, let the property $P(v)$ be the sentence
|
||||
|
||||
For every string $u$ in $S$, $uv$ is a string in $S$.
|
||||
|
||||
We will show that $P(v)$ is true for every string $v$ in $S$.
|
||||
|
||||
_Show that $P(a)$ is true for each string $a$ in the base for $S$:_
|
||||
|
||||
The only string in the base for $S$ is $\lambda$, and if $u$ is any string in
|
||||
$S$, then by rule II(b) in the definition of string, $u\lambda = u$. Hence the
|
||||
concatenation of $u$ and $\lambda$ is a string in $S$, and so $P(\lambda)$ is
|
||||
true.
|
||||
|
||||
_Show that for each string $x$ in $S$, if $P(x)$ is true and if $y$ is obtained
|
||||
from $x$ by applying a rule from the recursion for $S$, then $P(y)$ is true:_
|
||||
|
||||
The recursive definition for $S$ consists of three rules denoted II(a), II(b),
|
||||
and II(c\), but rule II(a) is the only one that generates new strings in $S$.
|
||||
Suppose $v$ is any string in $S$ such that $P(v)$ is true. In other words,
|
||||
suppose that for every string $u$ in $S$, $uv$ is a string in $S$. _[This is the
|
||||
inductive hypothesis.]_
|
||||
|
||||
Then rule II(a) is applied to $v$, the result is $vc$, where $c$ is a character
|
||||
in $A$. To complete the inductive step, we must show that $P(vc)$ is true. To do
|
||||
so, we will show that $u(vc)$ is a string in $S$.
|
||||
|
||||
Now because $uv$ is a string in $S$, it follows from rule II(a) that $(uv)c$ is
|
||||
also a string in $S$. In addition, by rule II\(c\),
|
||||
|
||||
$$ (uv)c = u(vc) $$
|
||||
|
||||
Therefore, $u(vc)$ is a string in $S$, which means that $P(vc)$ is true _[as was
|
||||
to be shown]_.
|
||||
|
||||
_Conclusion:_
|
||||
|
||||
Because there are no strings in $S$ other than those obtained from 5he base and
|
||||
the recursion for $S$, we conclude that the concatenation of any two strings in
|
||||
$S$ is a string in $S$.
|
||||
|
||||
---
|
||||
|
||||
Page 394
|
||||
|
||||
**Theorem 5.9.4 Concatenation of Strings is Associative**
|
||||
|
||||
If $S$ is the set of all strings over a finite set $A$ and $u$, $v$, and $w$ are
|
||||
any strings in $S$, then $u(vw) = (uv)w$.
|
||||
|
||||
**Idea of a 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 $S$, $u(vw) = (uv)w$.
|
||||
|
||||
The proof must show (1) that $P(\lambda)$ is true, and (2) that if $w$ is any
|
||||
string in $S$ such that $P(w)$ is true and if $y$ is obtained from $w$ by
|
||||
applying a rule from the recursion for $S$, then $P(y)$ is true. Now when rule
|
||||
II(a) is applied to $w$ the result is $wc$ for some character $c$ in $A$. A
|
||||
crucial step is to show that $u((vw)c) = (u(vw))c$. This follows from the
|
||||
definition of string because $u$ and $vw$ are in $S$ and $c$ is in $A$.
|
||||
|
||||
Exercise 21 at the end of this section asks you to write a complete proof.
|
||||
|
||||
---
|
||||
|
|
|
|||
|
|
@ -253,3 +253,25 @@ complex numbers.
|
|||
sequence is given by an explicit formula of the form _____.
|
||||
|
||||
$a_n = Cr^n + Dnr^n$ where $C$ and $D$ are real numbers.
|
||||
|
||||
---
|
||||
|
||||
Page 397
|
||||
|
||||
**Test Yourself**
|
||||
|
||||
1. The base for a recursive definition of a set is _____.
|
||||
|
||||
2. The recursion for a recursive definition of a set is _____.
|
||||
|
||||
3. The restriction for a recursive definition of a set is _____.
|
||||
|
||||
4. One way to show that a given element is in a recursively defined set is to
|
||||
start with an element or elements in the _____ and apply the rules from the
|
||||
_____ until you obtain the given element.
|
||||
|
||||
5. To use structural induction to prove that every element in a recursively
|
||||
defined set $S$ satisfies a certain property, you show that _____ and that,
|
||||
for each rule in the recursion, if _____ then _____.
|
||||
|
||||
6. A function is said to be defined recursively if, and only if, _____.
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue