🚧 Setup for 7.2
This commit is contained in:
tomit4
parent
b5b9cfae39
commit
b9536b5034
3 changed files with 595 additions and 0 deletions
|
|
@ -1330,3 +1330,405 @@ Omitted.
|
|||
perfect square.
|
||||
|
||||
Omitted.
|
||||
|
||||
---
|
||||
|
||||
Page 480
|
||||
|
||||
**Exercise Set 7.2**
|
||||
|
||||
1. The definition of one-to-one is stated in two ways:
|
||||
|
||||
$$ \forall x_1, x_2 \in X, \text{ if } F(x_1) = F(x_2) \tex{ then } x_1 = x_2 $$
|
||||
|
||||
and
|
||||
|
||||
$$ \forall x_1, x_2 \in X, \text{ if } x_1 \neq x_2 \tex{ then } F(x_1) \neq F(x_2) $$
|
||||
|
||||
Why are these two statements logically equivalent?
|
||||
|
||||
2. Fill in each blank with the word _most_ or _least_.
|
||||
|
||||
a. A function $F$ is one-to-one if, and only if, each element in the co-domain
|
||||
of $F$ is the image of at _____ one element in the domain of $F$.
|
||||
|
||||
b. A function $F$ is onto if, and only if, each element in the co-domain of $F$
|
||||
is the image of at _____ one element in the domain of $F$.
|
||||
|
||||
3. When asked to state the definition of one-to-one, a student replies, "A
|
||||
function $f$ is one-to-one if, and only if, every element of $X$ is sent by
|
||||
$f$ to exactly one element of $Y$." Give a counterexample to show that the
|
||||
student's reply is incorrect.
|
||||
|
||||
4. Let $f: X \to Y$ be a function. True or false? A sufficient condition for $f$
|
||||
to be one-to-one is that for every element $y$ in $Y$, there is at most one
|
||||
$x$ in $X$ with $f(x) = y$. Explain your answer.
|
||||
|
||||
5. All but two of the following statements are correct ways to express the fact
|
||||
that a function $f$ is onto. Find the two that are incorrect.
|
||||
|
||||
a. $f$ is onto $\Leftrightarrow$ every element in its co-domain is the image of
|
||||
some element in its domain.
|
||||
|
||||
b. $f$ is onto $\Leftrightarrow$ every element in its domain has a corresponding
|
||||
image in its co-domain.
|
||||
|
||||
c. $f$ is onto $\Leftrightarrow \forall y \in Y, \exists x \in X$ such that
|
||||
$f(x) = y$.
|
||||
|
||||
d. $f$ is onto $\Leftrightarrow \forall x \in X, \exists y \in Y$ such that
|
||||
$f(x) = y$.
|
||||
|
||||
e. $f$ is onto $\Leftrightarrow$ the range of $f$ is the same as the co-domain
|
||||
of $f$.
|
||||
|
||||
6. Let $X = \{1, 5, 9\}$ and $Y = \{3, 4, 7\}$.
|
||||
|
||||
a. Define $f: X \to Y$ by specifying that
|
||||
|
||||
$$ f(1) = 4, f(5) = 7, f(9) = 4 $$
|
||||
|
||||
Is $f$ one-to-one? Is $f$ onto? Explain your answers.
|
||||
|
||||
b. Define $g: X \to Y$ by specifying that
|
||||
|
||||
$$ g(1) = 7, g(5) = 3, g(9) = 4 $$
|
||||
|
||||
Is $g$ one-to-one? Is $g$ onto? Explain your answers.
|
||||
|
||||
7. Let $X = \{a, b, c, d\}$ and $Y = \{e, f, g\}$. Define functions $F$ and $G$
|
||||
by the arrow diagrams below.
|
||||
|
||||
(See page 481) for images.
|
||||
|
||||
a. Is $F$ one-to-one? Why or why not? Is it onto? Why or why not?
|
||||
|
||||
b. Is $G$ one-to-one? Why or why not? Is it onto? Why or why not?
|
||||
|
||||
8. Let $X = \{a, b, c\}$ and $Y = \{d, e, f, g\}$. Define functions $H$ and $K$
|
||||
by the arrow diagrams below.
|
||||
|
||||
(See page 481) for images.
|
||||
|
||||
a. Is $H$ one-to-one? Why or why not? Is it onto? Why or why not?
|
||||
|
||||
b. Is $K$ one-to-one? Why or why not? Is it onto? Why or why not?
|
||||
|
||||
9. Let $X = \{1, 2, 3\}$, $Y = \{1, 2, 3, 4\}$, and $Z = \{1, 2\}$.
|
||||
|
||||
a. Define a function $f: X \to Y$ that is one-to-one but not onto.
|
||||
|
||||
b. Define a function $g: X \to Z$ that is onto but not one-to-one.
|
||||
|
||||
c. Define a function $h: X \to X$ that is neither one-to-one nor onto.
|
||||
|
||||
d. Define a function $k: X \to X$ that is one-to-one and onto but is not the
|
||||
identity function on $X$.
|
||||
|
||||
10.
|
||||
|
||||
a. Define $f: \mathbb{Z} \to \mathbb{Z}$ by the rule $f(n) = 2n$, for every
|
||||
integer $n$.
|
||||
|
||||
i. Is $f$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
ii. Is $f$ onto? prove or give a counterexample.
|
||||
|
||||
b. Let $2\mathbb{Z}$ denote the set of all even integers. That is,
|
||||
$2\mathbb{Z} = \{n \in \mathbb{Z} | n = 2k \text{, for some integer } k\}$.
|
||||
Define $h: \mathbb{Z} \to 2\mathbb{Z}$ by the rule $h(n) = 2n$, for each integer
|
||||
$n$. Is $h$ onto? Prove or give a counterexample.
|
||||
|
||||
11.
|
||||
|
||||
a. Define $g: \mathbb{Z} \to \mathbb{Z}$ by the rule $g(n) = 4n - 5$, for each
|
||||
integer $n$.
|
||||
|
||||
i. Is $g$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
ii. Is $g$ onto? Prove or give a counterexample.
|
||||
|
||||
b. Define $G: \mathbb{R} \to \mathbb{R}$ by the rule $G(x) = 4x - 5$ for every
|
||||
real number $x$. Is $G$ onto? Prove or give a counterexample.
|
||||
|
||||
12.
|
||||
|
||||
a. Define $F: \mathbb{Z} \to \mathbb{Z}$ by the rule $F(n) = 2 - 3n$, for each
|
||||
integer $n$.
|
||||
|
||||
i. Is $F$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
ii. Is $F$ onto? Prove or give a counterexample.
|
||||
|
||||
b. Define $G: \mathbb{R} \to \mathbb{R}$ by the rule $G(x) = 2 - 3x$ for each
|
||||
real number $x$. Is $G$ onto? Prove or give a counterexample.
|
||||
|
||||
13.
|
||||
|
||||
a. Define $H: \mathbb{R} \to \mathbb{R}$ by the rule $H(x) = x^2$, for each real
|
||||
number $x$.
|
||||
|
||||
i. Is $H$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
ii. Is $H$ onto? Prove or give a counterexample.
|
||||
|
||||
b. Define $K: \mathbb{R}^{\text{nonneg}} \to \mathbb{R}^{\text{nonneg}}$ by the
|
||||
rule $K(x) = x^2$, for each nonnegative real number $x$. Is $K$ onto? Prove or
|
||||
give a counterexample.
|
||||
|
||||
14. Explain the mistake in the following "proof."
|
||||
|
||||
**Theorem:** The function $f: \mathbb{Z} \to \mathbb{Z}$ defined by the formula
|
||||
$f(n) = 4n + 3$, for each integer $n$, is one-to-one.
|
||||
|
||||
"**Proof:** Suppose any integer $n$ is given. Then by definition of $f$, there
|
||||
is only one possible value for $f(n)$ - namely, $4n + 3$. Hence $f$ is
|
||||
one-to-one."
|
||||
|
||||
In each of 15-18 a function $f$ is defined on a set of real numbers. Determine
|
||||
whether or not $f$ is one-to-one and justify your answer.
|
||||
|
||||
15. $f(x) = \dfrac{x + 1}{x}$, for each number $x \neq 0$
|
||||
|
||||
16. $f(x) = \dfrac{x}{x^2 + 1}$, for each real number $x$
|
||||
|
||||
17. $f(x) = \dfrac{3x - 1}{x}$, for each real number $x \neq 0$
|
||||
|
||||
18. $f(x) = \dfrac{x + 1}{x - 1}$, for each real number $x \neq 1$
|
||||
|
||||
19. Referring to Example 7.2.3, assume that records with the following ID
|
||||
numbers are to be placed in sequence into Table 7.2.1. Find the position
|
||||
into which each record is placed.
|
||||
|
||||
a. $417302072$
|
||||
|
||||
b. $364981703$
|
||||
|
||||
c. $283090787$
|
||||
|
||||
20. Define $\text{Floor}: \mathbb{R} \to \mathbb{Z}$ by the formula
|
||||
$\text{Floor}(x) = \lfloor x \rfloor$, for every real number $x$.
|
||||
|
||||
a. Is $\text{Floor}$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $\text{Floor}$ onto? Prove or give a counterexample.
|
||||
|
||||
21. Let $S$ be the set of all strings of $0$'s and $1$'s, and define
|
||||
$L: S \to \mathbb{Z}^{\text{nonneg}}$ by
|
||||
|
||||
$$ L(s) = \text{ the length of } s \text{, for every string } s \text{ in } S $$
|
||||
|
||||
a. Is $L$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $L$ onto? Prove or give a counterexample.
|
||||
|
||||
22. Let $S$ be the set of all strings of $0$'s and $1$'s, and define
|
||||
$D: S \to \mathbb{Z}$ as follows: For every $s \in S$,
|
||||
|
||||
$$ D(s) = \text{ the number of 1's in } s \text{ minus the number of 0's in } s $$
|
||||
|
||||
a. Is $D$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $D$ onto? Prove or give a counterexample.
|
||||
|
||||
23. Define $F: \mathscr{P}(\{a, b, c\}) \to \mathbb{Z}$ as follows: For every
|
||||
$A$ in $\mathscr{P}(\{a, b, c\})$,
|
||||
|
||||
$$ F(A) = \text{ the number of elements in } A $$
|
||||
|
||||
a. Is $F$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $F$ onto? Prove or give a counterexample.
|
||||
|
||||
24. Let $S$ be the set of all strings of $a$'s and $b$'s, and define
|
||||
$N: S \to \mathbb{Z}$ by
|
||||
|
||||
$$ N(s) = \text{ the number of a's in } s \text{, for each } s \in S $$
|
||||
|
||||
a. Is $N$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $N$ onto? Prove or give a counterexample.
|
||||
|
||||
25. Let $S$ be the set of all strings in $a$'s and $b$'s, and define
|
||||
$C: S \to S$ by
|
||||
|
||||
$$ C(s) = as \text{, for each } s \in S $$
|
||||
|
||||
($C$ is called **concatenation** by $a$ on the left.)
|
||||
|
||||
a. Is $C$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $C$ onto? Prove or give a counterexample.
|
||||
|
||||
26. Define $S: \mathbb{Z}^+ \to \mathbb{Z}^+$ by the rule: For each integer $n$,
|
||||
|
||||
$$ S(n) = \text{ the sum of the positive divisors of } n $$
|
||||
|
||||
a. Is $S$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $S$ onto? Prove or give a counterexample.
|
||||
|
||||
27. Let $D$ be the set of all finite subsets of positive integers, and define
|
||||
$T: \mathbb{Z}^+ \to D$ by the following rule:
|
||||
|
||||
For every integer $n$,
|
||||
$T(n) = \text{ the set of all of the positive divisors of } n$.
|
||||
|
||||
a. Is $T$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $T$ onto? Prove or give a counterexample.
|
||||
|
||||
28. Define $G: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ as
|
||||
follows:
|
||||
|
||||
$$ G(x, y) = (2y, -x) \text{ for every } (x, y) \in \mathbb{R} \times \mathbb{R} $$
|
||||
|
||||
a. Is $G$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $G$ onto? Prove or give a counterexample.
|
||||
|
||||
29. Define $H: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ as
|
||||
follows:
|
||||
|
||||
$$ H(x, y) = (x + 1, 2 - y) \text{ for every } (x, y) \in \mathbb{R} \times \mathbb{R} $$
|
||||
|
||||
a. Is $H$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $H$ onto? Prove or give a counterexample.
|
||||
|
||||
30. Define $J: \mathbb{Q} \times \mathbb{Q} \to \mathbb{R}$ by the rule
|
||||
|
||||
$$ J(r, s) = r + \sqrt{2}s \text{ for each } (r, s) \in \mathbb{Q} \times \mathbb{Q} $$
|
||||
|
||||
a. Is $J$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $J$ onto? Prove or give a counterexample.
|
||||
|
||||
31. Define $F: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+$ and
|
||||
$G: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+$ as follows:
|
||||
|
||||
For each $(n, m) \in \mathbb{Z}^+ \times \mathbb{Z}^+$,
|
||||
|
||||
$$ F(n, m) = 3^n5^m \text{ and } G(n, m) = 3^n6^m $$
|
||||
|
||||
a. Is $F$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
b. Is $G$ one-to-one? Prove or give a counterexample.
|
||||
|
||||
32.
|
||||
|
||||
a. Is $\log_{8}27 = \log_{2}3$? Why or why not?
|
||||
|
||||
a. Is $\log_{16}9 = \log_{4}3$? Why or why not?
|
||||
|
||||
The properties of logarithm established in 33-35 are used in Sections 11.4 and
|
||||
11.5.
|
||||
|
||||
33. Prove that for all positive real numbers $b$, $x$, and $y$ with $b \neq 1$,
|
||||
|
||||
$$ \log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y $$
|
||||
|
||||
34. Prove that for all positive real numbers $b$, $x$, and $y$ with $b \neq 1$,
|
||||
|
||||
$$ \log_{b}(xy) = \log_{b}x + \log_{b}y $$
|
||||
|
||||
35. Prove that for all real numbers $a$, $b$, and $x$ with $b$ and $x$ positive
|
||||
and $b \neq 1$,
|
||||
|
||||
$$ \log_{b}(x^a) = a\log_{b}x $$
|
||||
|
||||
Exercises 36 and 37 use the following definition: If
|
||||
$f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ are functions,
|
||||
then the function $(f + g): \mathbb{R} \to \mathbb{R}$ is defined by the formula
|
||||
$(f + g)(x) = f(x) + g(x)$ for every real number $x$.
|
||||
|
||||
36. If $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ are
|
||||
both one-to-one, is $f + g$ also one-to-one? Justify your answer.
|
||||
|
||||
37. If $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ are
|
||||
both onto, is $f + g$ also onto? Justify your answer.
|
||||
|
||||
Exercises 38 and 39 use the following definition: If
|
||||
$f: \mathbb{R} \to \mathbb{R}$ and $c$ is a nonzero real number, the function
|
||||
$(c \cdot f): \mathbb{R} \to \mathbb{R}$ is defined by the formula
|
||||
$(c \cdot f)(x) = c \cdot (f(x))$ for every real number $x$.
|
||||
|
||||
38. Let $f: \mathbb{R} \to \mathbb{R}$ be a function and $c$ a nonzero real
|
||||
number. If $f$ is one-to-one, is $c \cdot f$ also one-to-one? Justify your
|
||||
answer.
|
||||
|
||||
39. Let $f: \mathbb{R} \to \mathbb{R}$ be a function and $c$ a nonzero real
|
||||
number. If $f$ is onto, is $c \cdot f$ also onto? Justify your answer.
|
||||
|
||||
40. Suppose $F: X \to Y$ is one-to-one.
|
||||
|
||||
a. Prove that for every subset $A \subseteq X$, $F^{-1}(F(A)) = A$.
|
||||
|
||||
b. Prove that for all subsets $A_1$ and $A_2$ in $X$,
|
||||
$F(A_1 \cap A_2) = F(A_1) \cap F(A_2)$.
|
||||
|
||||
Let $X = \{a, b, c, d, e\}$ and $Y = \{s, t, u, v, w\}$. In each of 42 and 43 a
|
||||
one-to-one correspondence $F: X \to Y$ is defined by an arrow diagram. In each
|
||||
case draw an arrow diagram for $F^{-1}$.
|
||||
|
||||
42.
|
||||
|
||||
(See page 483 for image.)
|
||||
|
||||
43.
|
||||
|
||||
(See page 483 for image.)
|
||||
|
||||
In 44-55 indicate which of the functions in the referenced exercise are
|
||||
one-to-one correspondences. For each function that is a one-to-one
|
||||
correspondence, find the inverse function.
|
||||
|
||||
44. Exercise 10a
|
||||
|
||||
45. Exercise 10b
|
||||
|
||||
46. Exercise 11a
|
||||
|
||||
47. Exercise 11b
|
||||
|
||||
48. Exercise 12a
|
||||
|
||||
49. Exercise 12b
|
||||
|
||||
50. Exercise 21
|
||||
|
||||
51. Exercise 22
|
||||
|
||||
52. Exercise 15 with the co-domain taken to be the set of all real numbers not
|
||||
equal to $1$.
|
||||
|
||||
53. Exercise 16 with the co-domain taken to be the set of all real numbers.
|
||||
|
||||
54. Exercise 17 with the co-domain taken to be the set of all real numbers not
|
||||
equal to $3$
|
||||
|
||||
55. Exercise 18 with the co-domain taken to be the set of all real numbers not
|
||||
equal to 1.
|
||||
|
||||
56. In Example 7.2.8 a one-to-one correspondence was defined from the power set
|
||||
of $\{a, b\}$ to the set of all strings of $0$'s and $1$'s that have length
|
||||
$2$. Thus the elements of these two sets can be matched up exactly, and so
|
||||
the two sets have the same number of elements.
|
||||
|
||||
a. Let $X = \{x_1, x_2, \dots, x_n\}$ be a set with $n$ elements. Use Example
|
||||
7.2.8 as a model to define a one-to-one correspondence from $\mathscr{P}(X)$,
|
||||
the set of all subsets of $X$, to the set of all strings of $0$'s and $1$'s that
|
||||
have length $n$.
|
||||
|
||||
b. In Section 9.2 we show that there are $2^n$ strings of $0's$ and $1$'s that
|
||||
have length $n$. What does this allow you to conclude about the number of
|
||||
subsets of $\mathscr{P}(X)$? (This provides an alternative proof of Theorem
|
||||
6.3.1.)
|
||||
|
||||
57. Write a computer algorithm to check whether a function from one finite set
|
||||
to another is one-to-one. Assume the existence of an independent algorithm
|
||||
to compute values of the function.
|
||||
|
||||
58. Write a computer algorithm to check whether a function from one finite set
|
||||
to another is onto. Assume the existence of an independent algorithm to
|
||||
compute values of the function.
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue