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Page 458
**Exercise Set 7.1**
1. Let $X = \{1, 3, 5\}$ and $Y = \{s, t, u, v\}$. Define $f: X \to Y$ by the
following arrow diagram.
(See page 458 for image)
a. Write the domain of $f$ and the co-domain of $f$.
b. Find $f(1)$, $f(3)$, and $f(5)$.
c. What is the range of $f$?
d. Is $3$ an inverse image of $s$? Is $1$ an inverse image of $u$?
e. What is the inverse image of $s$? of $u$? of $v$?
f. Represent $f$ as a set of ordered pairs.
2. Let $X = \{1, 3, 5\}$ and $Y = \{a, b, c, d\}$. Define $g: X \to Y$ by the
following arrow diagram.
(See page 459 for image)
a. Write the domain of $g$ and the co-domain of $g$.
b. Find $g(1)$, $g(3)$, and $g(5)$.
c. What is the range of $g$?
d. Is $3$ an inverse image of $a$? Is $1$ an inverse image of $b$?
e. What is the inverse image of $b$? of $c$?
f. Represent $g$ as a set of ordered pairs.
3. Indicate whether the statements in parts (a)-(d) are true or false for all
functions. Justify your answers.
a. If two elements in the domain of a function are equal, then their images in
the co-domain are equal.
b. If two elements in the co-domain of a function are equal, then their
preimages in the domain are also equal.
c. A function can have the same output for more than one input.
d. A function can have the same input for more than one output.
4.
a. Find all functions from $X = \{a, b\}$ to $Y = \{u, v\}$.
b. Find all functions from $X = \{a, b, c\}$ to $Y = \{u\}$.
c. Find all functions from $X = \{a, b, c\}$ to $Y = \{u, v\}$.
5. Let $I_{\mathbb{z}}$ bee the identity function defined on the set of all
integers, and suppose that $e$, $b_i^{jk}$, $K(t)$, and $u_{kj}$ all
represent integers. Find the following:
a. $I_{\mathbb{Z}}(e)$
b. $I_{\mathbb{Z}}\left(b_i^{jk}\right)$
c. $I_{\mathbb{Z}}(K(t))$
d. $I_{\mathbb{Z}}(u_{kj})$
6. Find functions defined on the set of nonnegative integers that can be used to
define the sequences whose first six terms are given below.
a. $1, -\dfrac{1}{3}, \dfrac{1}{5}, -\dfrac{1}{7}, \dfrac{1}{9}, -\dfrac{1}{11}$
b. $0, -2, 4, -6, 8, -10$
7. Let $A = \{1, 2, 3, 4, 5\}$, and define a function
$F: \mathscr{P}(A) \to \mathbb{Z}$ as follows: For each set $X$ in
$\mathscr{P}(A)$,
$$
F(x) =
\begin{cases}
0& \text{if } X \text{ has an even number of elements} \\
1 & \text{if } X \text{ has an odd number of elements}
\end{cases}
$$
Find the following:
a. $F(\{1, 3, 4\})$
b. $F(\emptyset)$
c. $F(\{2, 3\})$
d. $F(\{2, 3, 4, 5\})$
8. Let $J_5 = \{0, 1, 2, 3, 4\}$, and define a function $F: J_5 \to J_5$ as
follows: For each $x \in J_5$, $F(x) = (x^3 + 2x + 4) \mod 5$.
Find the following:
a. $F(0)$
b. $F(1)$
c. $F(2)$
d. $F(3)$
e. $F(4)$
9. Define a function $S: \mathbb{Z}^+ \to \mathbb{Z}^+$ as follows: For each
positive integer $n$,
$$ S(n) = \text{ the sum of the positive divisors of } n $$
Find the following:
a. $S(1)$
b. $S(15)$
c. $S(17)$
d. $S(5)$
e. $S(18)$
f. $S(21)$
10. Let $D$ be the set of all finite subsets of positive integers.
Define a function $T: \mathbb{Z}^+ \to D$ as follows: For each positive integer
$n$, $T(n) =$ the set of positive divisors of $n$.
Find the following:
a. $T(1)$
b. $T(15)$
c. $T(17)$
d. $T(5)$
e. $T(18)$
f. $T(21)$
11. Define $F: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ as
follows: For every ordered pair $(a, b)$ of integers,
$F(a, b) = (2a + 1, 3b - 2)$.
Find the following:
a. $F(4, 4)$
b. $F(2, 1)$
c. $F(3, 2)$
d. $F(1, 5)$
12. Let $J_5 = \{0, 1, 2, 3, 4\}$, and define
$G: J_5 \times J_5 \to J_5 \times J_5$ as follows: For each
$(a, b) \in J_5 \times J_5$,
$$ G(a, b) = ((2a + 1) \mod 5, (3b - 2) \mod 5) $$
Find the following:
a. $G(4, 4)$
b. $G(2, 1)$
c. $G(3, 2)$
d. $G(1, 5)$
13. Let $J_5 = \{0, 1, 2, 3, 4\}$, and define functions $f: J_5 \to J_5$ and
$g: J_5 \to J_5$ as follows: For each $x \in J_5$,
$$ f(x) = (x + 4)^2 \mod 5 \quad \text{ and } \quad g(x) = (x^2 + 3x + 1) \mod 5 $$
Is $f = g$? Explain.
14. Define functions $H$ and $K$ from $\mathbb{R}$ to $\mathbb{R}$ by the
following formulas:
For every $x \in \mathbb{R}$,
$$ H(x) = \lfloor x \rfloor + 1 \quad \text{ and } \quad K(x) = \lceil x \rceil $$
Does $H = K$? Explain.
15. Let $F$ and $G$ be functions from the set of all real numbers to itself.
Define the product functions $F \cdot G: \mathbb{R} \to \mathbb{R}$ and
$G \cdot F: \mathbb{R} \to \mathbb{R}$ as follows: For every
$x \in \mathbb{R}$,
$$ (F \cdot G)(x) = F(x) \cdot G(x) $$
$$ (G \cdot F)(x) = G(x) \cdot F(x) $$
Does $F \cdot G = G \cdot F$? Explain.
16. Let $F$ and $G$ be function sfrom the set of all real numbers to itself.
Define new functions $F - G: \mathbb{R} \to \mathbb{R}$ and
$G - F: \mathbb{R} \to \mathbb{R}$ as follows: For every $x \in \mathbb{R}$,
$$ (F - G)(x) = F(x) - G(x) $$
$$ (G - F)(x) = G(x) - F(x) $$
Does $F - G = G - F$? Explain.
17. Use the definition of logarithm to fill in the blanks below.
a. $\log_28 = 3$ because _____.
b. $\log_5\left(\dfrac{1}{25}\right) = -2$ because _____.
c. $\log_44 = 1$ because _____.
d. $\log_3(3^n) = n$ because _____.
e. $\log_41 = 0$ because _____.
18. Find exact values for each of the following quantities without using a
calculator.
a. $\log_{3}81$
b. $\log_{2}1024$
c. $\log_{3}\left(\dfrac{1}{27}\right)$
d. $\log_{2}1$
e. $\log_{10}\left(\dfrac{1}{10}\right)$
f. $\log_{3}3$
g. $\log_{2}(2^k)$
19. Use the definition of logarithm to prove that for any positive real number
$b$ with $b \neq 1$, $\log_{b}b = 1$.
20. Use the definition of logarithm to prove that for any positive real number
$b$ with $b \neq 1$, $\log_{b}1 = 0$.
21. If $b$ is any positive real number with $b \neq 1$ and $x$ is any real
number, $b^{-x}$ is defined as follows:
$b^{-x} = \dfrac{1}{b^x}$. Use this definition and the definition of logarithm
to prove that $\log_{b}\left(\dfrac{1}{u}\right) = -\log_{b}u$ for all positive
real numbers $u$ and $b$, with $b \neq 1$.
22. Use the unique factorization for the integers theorem (Section 4.4) and the
definition of logarithm to prove that $\log_{3}(7)$ is irrational.
23. If $b$ and $y$ are positive real numbers such that $\log_{b}y = 3$, what is
$\log_{\frac{1}{b}}y$? Explain.
24. If $b$ and $y$ are positive real numbers such that $\log_{b}y = 2$, what is
$\log_{b^2}(y)$? Explain.
25. Let $A = \{2, 3, 5\}$ and $B = \{x, y\}$. Let $p_1$ and $p_2$ be the
**projections of $A \times B$ onto the first and second coordinates.** That
is, for each pair $(a, b) \in A \times B$, $p_1(a, b) = a$ and
$p_2(a, b) = b$.
a. Find $p_1(2, y)$ and $p_1(5, x)$. What is the range of $p_1$?
b. Find $p_2(2, y)$ and $p_2(5, x)$. What is the range of $p_2$?
26. Observe that $\mod$ and $\text{div}$ can be defined as functions from
$\mathbb{Z}^{\text{nonneg}}$ \times \mathbb{Z}^+$ to $\mathbb{Z}$. For each
ordered pair $(n, d)$ consisting of a nonnegative integer $n$ and a positive
integer $d$, let
$\mod(n, d) = n \mod d$ (the nonnegative remainder obtained when $n$ is divided
by $d$).
$\text{div}(n, d) = n \text{ div } d$ (the integer quotient obtained when $n$ is
divided by $d$).
Find each of the following:
a. $\mod(67, 10)$ and $\text{div}(67, 10)$
b. $\mod(59, 8)$ and $\text{div}(59, 8)$
c. $\mod(30, 5)$ and $\text{div}(30, 5)$
27. Let $S$ be the set of all strings of $a$'s and $b$'s.
a. Define $f: S \to Z$ as follows: For each string $s$ in $S$
$$
f(s) =
\begin{cases}
& \text{ the number of b's to the left-most a in s} \\
0 & \text{if s contains no a's}
\end{cases}
$$
Find $f(aba)$, $f(bbab)$, and $f(b)$. What is the range of $f$?
b. Define $g: S \to S$ as follows: For each string $s$ in $S$,
$$ g(s) = \text{ the string obtained by writing the characters of s in reverse order} $$
Find $g(aba)$, $g(bbab)$, and $g(b)$. What is the range of $g$?
28. Consider the coding and decoding functions $E$ and $D$ defined in Example
7.1.9.
a. Find $E(0110)$ and $D(111111000111)$.
b. Find $E(1010)$ and $D(000000111111)$.
29. Consider the Hamming distance function defined in Example 7.1.10.
a. Find $H(10101, 00011$.
b. Find $H(00110, 10111)$.
30. Draw arrow diagrams for the Boolean functions defined by the following
input/output tables.
a.
| Input | Intput | Output |
| ------- | ------ | ------ |
| $P$ | $Q$ | $R$ |
| ------- | - | |
| 1 | 1 | 0 |
| 1 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
b.
| Input | Intput | Input | Output |
| ----- | ------ | ----- | ------ |
| $P$ | $Q$ | $R$ | $S$ |
| - | - | - | - |
| 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
31. Fill in the following table to show the values of all possible two-place
Boolean functions.
| Input | Input | $f_1$ | $f_2$ | $f_3$ | $f_4$ | $f_5$ | $f_6$ | $f_7$ | $f_8$ | $f_9$ | $f_{10}$ | $f_{11}$ | $f_{12}$ | $f_{13}$ | $f_{14}$ | $f_{15}$ | $f_{16}$ |
| ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | -------- | -------- | -------- | -------- | -------- | -------- | -------- |
| 1 | 1 | | | | | | | | | | | | | | | | |
| 1 | 0 | | | | | | | | | | | | | | | | |
| 0 | 1 | | | | | | | | | | | | | | | | |
| 0 | 0 | | | | | | | | | | | | | | | | |
32. Consider the three-place Boolean function $f$ defined by the following rule:
For each triple $(x_1, x_2, x_3)$ of $0$'s and $1$'s,
$$ f(x_1, x_2, x_3) = (4x_1 + 3x_2 + 2x_3) \mod 2 $$
a. Find $f(1, 1, 1)$ and $f(0, 0, 1)$.
b. Describe $f$ using an input/output table.
33. Student A tries to define a function $g: \mathbb{Q} \to \mathbb{Z}$ by the
rule
$g\left(\dfrac{m}{n}\right) = m - n$, for all integers $m$ and $n$ with
$n \neq 0$.
Student B claims that $g$ is not well defined. Justify student B's claim.
34. Student C tries to define a function $h: \mathbb{Q} \to \mathbb{Q}$ by the
rule
$h\left(\dfrac{m}{n}\right) = \dfrac{m^2}{n}$, for all integers $m$ and $n$ with
$n \neq 0$.
Student D claims that $h$ is not well defined. Justify student D's claim.
35. Let $U = \{1, 2, 3, 4\}$. Student A tries to define a function $R: U \to Z$
as follows: For each $x \in U$,
$R(x)$ is the integer $y$ so that $(xy) \mod 5 = 1$.
Student B claims that $R$ is not well defined. Who is correct: student A or
student B? Justify your answer.
36. Let $V = \{1, 2, 3\}$. Student C tries to define a function $S: V \to V$ as
follows: For each $x \in V$,
$S(x)$ is the integer $y$ in $V$ so that $(xy) \mod 4 = 1$.
Student D claims that $S$ is not well defined. Who is right: student C or
student D? Justify your answer.
37. On certain computers the integer data type goes from $-2,147,483,648$
through $2,147,483,647$. Let $S$ be the set of all integers from
$-2,147,483,648$ through $2,147,483,647$. Try to define a function
$f: S \to S$ by the rule $f(n) = n^2$ for each $n$ in $S$. Is $f$ well
defined? Explain.
38. Let $X = \{a, b, c\}$ and $Y = \{r, s, t, u, v, w\}$. Define $f: X \to Y$ as
follows: $f(a) = v$, $f(b) = v$, and $f(c) = t$.
a. Draw an arrow diagram for $f$.
b. Let $A = \{a, b\}$, $C = \{t\}$, $D = \{u, v\}$, and $E = \{r, s\}$. Find
$f(A)$, $f(X)$, $f^{-1}(C)$, $f^{-1}(D)$, $f^{-1}(E)$, and $f^{-1}(Y)$.
39. Let $X = \{1, 2, 3, 4\}$ and $Y = \{a, b, c, d, e\}$. Define $g: X \to Y$ as
follows: $g(1) = a$, $g(2) = a$, $g(3) = a$, and $g(4) = d$.
a. Draw an arrow diagram for $g$.
b. Let $A = \{2, 3\}$, $C = \{a\}$, and $D = \{b, c\}$. Find $g(A)$, $g(X)$,
$g^{-1}(C)$, $g^{-1}(D)$, and $g^{-1}(Y)$.
40. Let $X$ and $Y$ be sets, let $A$ and $B$ be any subsets of $X$, and let $F$
be a function from $X$ to $Y$. Fill in the blanks in the following proof
that $F(A) \cup F(B) \subseteq F(A \cup B)$.
**Proof:**
Let $y$ be any element in $F(A) \cup F(B)$. _[We must show that $y$ is in
$F(A \cup B)$.]_ By definition of union, __ (i) __.
_Case 1 $y \in F(A)$:_
In this case, by definition of $F(A)$, $y = F(x)$ for __ (ii) __ $x \in A$.
Since $A \subseteq A \cup B$, it follows from the definition of union that
$x \in$ __ (iii) __. Hence, $y = F(x)$ for some $x \in A \cup B$, and thus, by
definition of $F(A \cup B)$, $y \in$ __ (iv) __.
_Case 2, $y \in F(B)$:_
In this case, by definition of $F(B)$, __ (v) __ for some $x \in B$. Since
$B \subseteq A \cup B$ it follows from the definition of union that __ (vi) __.
Thus $y \in F(A \cup B)$.
Therefore, regardless of whether $y \in F(A)$ or $y \in F(B)$, we have that
$y \in F(A \cup B)$ _[as was to be shown]_.
In 41-49 let $X$ and $Y$ be sets, let $A$ and $B$ be any subsets of $X$, and let
$C$ and $D$ be any subsets of $Y$. Determine which of the properties are true
for every function $F$ from $X$ to $Y$ and which are false for at least one
function $F$ from $X$ to $Y$. Justify your answers.
41. If $A \subseteq B$ then $F(A) \subseteq F(B)$
42. $F(A \cap B) \subseteq F(A) \cap F(B)$
43. $F(A) \cap F(B) \subseteq F(A \cap B)$
44. For all subsets $A$ and $B$ of $X$, $F(A - B) = F(A) - F(B)$.
45. For all subsets $C$ and $D$ of $Y$, if $C \subseteq D$, then
$F^{-1}(C) \subseteq F^{-1}(D)$.
46. For all subsets $C$ and $D$ of $Y$,
$$ F^{-1}(C \cup D) = F^{-1}(C) \cup F^{-1}(D) $$
47. For all subsets $C$ and $D$ of $Y$,
$$ F^{-1}(C \cap D) = F^{-1}(C) \cap F^{-1}(D) $$
48. For all subsets $C$ and $D$ of $Y$,
$$ F^{-1}(C - D) = F^{-1}(C) - F^{-1}(D) $$
49. $F(F^{-1}(C)) \subseteq C$
50. Given a set $S$ and a subset $A$, the **characteristic function of $A$**,
denoted $\chi_A$, is the function defined from $S$ to $\mathbb{Z}$ with the
property that for each $u \in S$,
$$
\chi_{A}(u) =
\begin{cases}
1 & \text{if } u \in A \\
0 & \text{if } u \notin A
\end{cases}
$$
Show that each of the following holds for all subsets $A$ and $B$ of $S$ and
every $u \in S$.
a. \chi_{A \cap B}(u) = \chi_{A}(u) \cdot \chi_{B}(u)
b.
$\chi_{A \cup B}(u) = \chi_{A}(u) + \chi_{B}(u) - \chi_{A}(u) \cdot \chi_{B}(u)$
Each of exercises 51-53 refers to the Euler phi function, denoted $\phi$, which
is defined as follows: For each integer $n \geq 1$, $\phi(n)$ is the number of
positive integers less than or equal to $n$ that have no common factors with $n$
except $\pm 1$. For example $\phi(10) = 4$ because there are four positive
integers less than or equal to $10$ that have no common factors with $10$ except
$\pm 1$ - namely, $1$, $3$, $7$, and $9$.
51. Find each of the following:
a. $\phi(15)$
b. $\phi(2)$
c. $\phi(5)$
d. $\phi(12)$
e. $\phi(11)$
f. $\phi(1)$
52. Prove that if $p$ is a prime number and $n$ is an integer with $n \geq 1$,
then $\phi(p^n) = p^n - p^{n - 1}$.
53. Prove that there are infinitely many integers $n$ for which $\phi(n)$ is a
perfect square.

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Page 449
**Definition**
A **function $f$ from a set $X$ to a set $Y$**, denoted: $f: X \to Y$, is a
relation from $X$, the **domain** of $f$, to $Y$, the **co-domain** of $f$, that
satisfies two properties: (1) every element in $X$ is related to some element in
$Y$, and (2) no element in $X$ is related to more than one element in $Y$. Thus,
given any element $x$ in $X$, there is a unique element in $Y$ that is related
to $x$ by $f$. If we call this element $y$, then we say that "$f$ sends $x$ to
$y$" or "$f$ maps $x$ to $y$" and write $x \xrightarrow{f} y$ or $f: x \to y$.
The unique element to which $f$ sends $x$ is denoted
$f(x)$ and is called $f$ of $x$, or the output of $f$ for the input $x$, or the
value of $f$ at $x$, or the image of $x$ under $f$.
The set of all values of $f$ taken together is called the _range of $f$_ or the
_image of $X$ under $f$_. Symbolically:
$$ \text{range of } f = \text{ image of } X \text{ under } f = \{y \in Y | y = f(x), \text{ for some } x \text{ in } X\} $$
Given an element $y$ in $Y$, there may exist elements in $X$ with $y$ as their
image. When $x$ is an element such that $f(x) = y$, then $x$ is called **a
preimage of $y$** or **an inverse image of $y$**. The set of all inverse images
of $y$ is called _the inverse image of $y$_. Symbolically:
$$ \text{ the inverse image of } y = \{x \in X | f(x) = y\} $$
---
Page 451
**Theorem 7.1.1 A Test for Function Equality**
If $F: X \to Y$ and $G: X \to Y$ are functions, then $F = G$ if, and only if,
$F(x) = G(x)$ for every $x \in X$.
**Proof:**
Suppose $F: X \to Y$ and $G: X \to Y$ are functions; that is, $F$ and $G$ are
relations from $X$ to $Y$ that satisfy the two additional function properties.
Then $F$ and $G$ are subsets of $X \times Y$, and for $(x, y)$ to be in $F$
means that $y$ is the unique element related to $x$ by $F$, which we denote as
$F(x)$. Similarly, for $(x, y)$ to be in $G$ means that $y$ is the unique
element related to $x$ by $G$, which we denote as $G(x)$.
Now suppose that $F(x) = G(x)$ for every $x \in X$. Then if $x$ is any element
of $X$,
$$ (x, y) \in F \Leftrightarrow y = F(x) \Leftrightarrow y = G(x) \Leftrightarrow (x, y) \in G $$
because $F(x) = G(x)$.
So $F$ and $G$ consist of exactly the same elements and hence $F = G$.
Conversely, if $F = G$, then for every $x \in X$,
$$ y = F(x) \Leftrightarrow (x, y) \in F \Leftrightarrow (x, y) \in G \Leftrightarrow y = G(x) $$
because $F$ and $G$ consist of exactly the same elements.
Thus, since both $F(x)$ and $G(x)$ equal $y$, we have that
$$ F(x) = G(x) $$
---
Page 453
**Definition Logarithms and Logarithmic Functions**
Let $b$ be a positive real number with $b \neq 1$. For each positive real number
$x$, the **logarithm with base $b$ of $x$**, written $\log_bx$, is the exponent
to which $b$ must be raised to obtain $x$. Symbolically:
$$ \log_bx = y \Leftrightarrow b^y = x $$
The **logarithmic function with base $b$** is the function from $\mathbb{R}^+$
to $\mathbb{R}$ that takes each positive real number $x$ to $\log_bx$.
---
Page 455
**Definition**
An **($n$-place) Boolean function** $f$ is a function whose domain is the set of
all ordered $n$-tuples of $0$'s and $1$'s and whose co-domain is the set
$\{0, 1\}$. More formally, the domain of a Boolean function can be described as
the Cartesian product of $n$ copies of the set $\{0, 1\}$, which is denoted
$\{0, 1\^n}$. Thus $f: \{0, 1\}^n \to \{0, 1\}$.
---
Page 457
**Definition**
If $f: X \to Y$ is a function and $A \subseteq X$ and $C \subseteq Y$, then
$$ f(A) = \{y \in Y | y = f(x) \text{ for some } x \text{ in } A\} $$
and
$$ f^{-1}(C) = \{x \in X | f(x) \in C\} $$
$f(A)$ is called the **image of $A$**, and $f^{-1}(C)$ is called the **inverse
image of $C$**.

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**Test Yourself**
1. Given a function $f$ from a set $X$ to a set $Y$, $f(x)$ is _____.
2. Given a function $f$ from a set $X$ to a set $Y$, if $f(x) = y$ then $y$ is
called _____ or _____ or _____.
3. Given a function $f$ from a set $X$ to a set $Y$, the range of $f$ (or the
image of $X$ under $f$) is _____.
4. Given a function $f$ from a set $X$ to $Y$, if $f(x) = y$ then $x$ is called
_____ or _____.
5. Given a function $f$ from a set $X$ to a set $Y$, if $y \in Y$ then
$f^{-1}(y) =$ _____ and is called _____.
6. Given functions $f$ and $g$ from a set $X$ to a set $Y$, $f = g$ if, and only
if, _____.
7. Given positive real numbers $x$ and $b$ with $b \neq 1$, $\log_b(x) =$ _____.
8. Given a function $f$ from a set $X$ to a set $Y$ and a subset $A$ of $X$,
$f(A) =$ _____.
9. Given a function $f$ from a set $X$ to a set $Y$ and a subset $C$ of $Y$,
$f^{-1}(C) =$ _____.