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chapter_7/exercises.md
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perfect square.
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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.

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chapter_7/notes.md
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@ -106,3 +106,155 @@ $$ f^{-1}(C) = \{x \in X | f(x) \in C\} $$
$f(A)$ is called the **image of $A$**, and $f^{-1}(C)$ is called the **inverse
image of $C$**.
---
Page 463
**Definition**
Let $F$ be a function from a set $X$ to a set $Y$. $F$ is **one-to-one** (or
**injective**) if, and only if, for all elements $x_1$ and $x_2$ in $X$,
$$ \text{if } F(x_1) = F(x_2) \text{, then } x_1 = x_2 $$
or, equivalently,
$$ \text{if } x_1 \neq x_2 \text{, then } F(x_1) \neq F(x_2) $$
Symbolically:
$$ F: X \to Y \text{ is one-to-one } \Leftrightarrow \forall x_1, x_2 \in X \text{, if } F(x_1) = F(x_2) \text{ then } x_1 = x_2 $$
---
Page 466
**Definition: Hash Function**
A **hash function** is a function defined from a larger, possibly infinite, set
of data to a smaller fixed-size set of integers.
---
Page 469
**Definition**
Let $F$ be a function from a set $X$ to a set $Y$. $F$ is **onto** (or
**surjective**) if, and only if, given any element $y$ in $Y$, it is possible to
find an element $x$ in $X$ with the property that $y = F(x)$.
Symbolically:
$$ F:X \to Y \text{ is onto } \Leftrightarrow \forall y \in Y, \exists x \in X \text{ such that } F(x) = y $$
---
Page 472
**Laws of Exponents**
If $b$ and $c$ are any positive real numbers and $u$ and $v$ are any real
numbers, the following laws of exponents hold true:
7.2.1
$$ b^ub^v = b^{u + v} $$
7.2.2
$$ (b^u)^v = b^{uv} $$
7.2.3
$$ \frac{b^u}{b^v} = b^{u - v} $$
7.2.4
$$ (bc)^u = b^uc^u $$
---
Page 473
**Theorem 7.2.1 Properties of Logarithms**
For any positive real numbers $b$, $c$, $x$ and $y$ with $b \neq 1$ and
$c \neq 1$ and for every real number $a$:
a. $\log_b(xy) = \log_bx + \log_by$
b. $\log_b\left(\dfrac{x}{y}\right) = \log_bx - \log_by$
c. $\log_b(x^a) = a\log_bx$
d. $\log_cx = \dfrac{\log_bx}{\log_bc}$
---
Page 475
**Definition**
A **one-to-one correspondence** (or **bijection**) from a set $X$ to a set $Y$
is a function $F: X \to Y$ that is both one-to-one and onto.
---
Page 478
**Theorem 7.2.2**
Suppose $F: X \to Y$ is a one-to-one correspondence; in other words, suppose $F$
is one-to-one and onto. Then there is a function $F^{-1}: Y \to X$ that is
defined as follows:
Given any element $y$ in $Y$,
$$ F^{-1}(y) = \text{ that unique element } x \text{ in } X \text{ such that } F(x) \text{ equals } y $$
Or, equivalently,
$$ F^{-1}(y) = x \Leftrightarrow y = F(x) $$
---
Page 478
**Definition**
The function $F^{-1}$ of Theorem 7.2.2 is called the **inverse function** for
$F$.
---
Page 479
**Theorem 7.2.3**
If $X$ and $Y$ are sets and $F: X \to Y$ is one-to-one and onto, then
$F^{-1}:Y \to X$ is also one-to-one and onto.
**Proof:**
**$F^{-1}$ is one-to-one:**
Suppose $y_1$ and $y_2$ are elements of $Y$ such that
$F^{-1}(y_1) = F^{-1}(y_2)$. _[We must show that $y_1 = y_2$.]_ Let
$x = F^{-1}(y_1) = F^{-1}(y_2)$. Then $x \in X$, and by definition of $F^{-1}$,
$$ F(x) = y_1 \text{ since } x = F^{-1}(y_1) $$
and
$$ F(x) = y^2 \text{ since } x = F^{-1}(y_2) $$
Consequently, $y_1 = y_2$ because each is equal to $F(x)$. _[This is what was to
be shown.]_
**$F^{-1}$ is onto:**
Suppose $x \in X$. _[We must show that there exists an element $y$ in $Y$ such
that $F^{-1}(y) = x$.]_ Let $y = F(x)$. Then $y \in Y$, and by definition of
$F^{-1}$, $F^{-1}(y) = x$ _[as was to be shown.]_

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chapter_7/test_yourself.md
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@ -45,3 +45,44 @@ $\{y \in Y | y = f(x) \text{ for some } x \in A\}$
$f^{-1}(C) =$ _____.
$\{x \in X | f(x) \in C\}$
---
Page 480
**Test Yourself**
1. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is one-to-one if,
and only if, _____.
2. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is not one-to-one
if, and only if, _____.
3. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is onto if, and
only if, _____.
4. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is not onto if,
and only if, _____.
5. The following two statements are _____:
$$ \forall u, v \in U, \text{ if } H(u) = H(v) \text{ then } u = v $$
$$ \forall u, v \in U, \text{ if } u \neq v \text{ then } H(u) \neq H(v) $$
6. Given a function $F: X \to Y$ where $X$ is an infinite set, to prove that $F$
is one-to-one, you suppose that _____ and then you show that _____.
7. Given a function $F: X \to Y$ where $X$ is an infinite set, to prove that $F$
is onto, you suppose that _____ and then you show that _____.
8. Given a function $F: X \to Y$, to prove that $F$ is not one-to-one, you
_____.
9. Given a function $F: X \to Y$, to prove that $F$ is not onto, you _____.
10. A one-to-one correspondence from a set $X$ to a st $Y$ is a _____ that is
_____.
11. If $F$ is a one-to-one correspondence from a set $X$ to a set $Y$ and $y$ is
in $Y$, then $F^{-1}(y)$ is _____.