From 99f4f289e30cb8cbf6d91068487b49bce5c54fa1 Mon Sep 17 00:00:00 2001 From: tomit4 Date: Sat, 25 Jul 2026 19:42:03 -0700 Subject: [PATCH] :construction: Setup for 7.1 --- chapter_7/exercises.md | 535 +++++++++++++++++++++++++++++++++++++ chapter_7/notes.md | 108 ++++++++ chapter_7/test_yourself.md | 28 ++ 3 files changed, 671 insertions(+) create mode 100644 chapter_7/exercises.md create mode 100644 chapter_7/notes.md create mode 100644 chapter_7/test_yourself.md diff --git a/chapter_7/exercises.md b/chapter_7/exercises.md new file mode 100644 index 0000000..2c3f30f --- /dev/null +++ b/chapter_7/exercises.md @@ -0,0 +1,535 @@ +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. diff --git a/chapter_7/notes.md b/chapter_7/notes.md new file mode 100644 index 0000000..bae55fe --- /dev/null +++ b/chapter_7/notes.md @@ -0,0 +1,108 @@ +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 $C$**. diff --git a/chapter_7/test_yourself.md b/chapter_7/test_yourself.md new file mode 100644 index 0000000..681359d --- /dev/null +++ b/chapter_7/test_yourself.md @@ -0,0 +1,28 @@ +Page 458 + +**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 $Y$, + $f^{-1}(C) =$ _____.