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.