From b9536b5034003e146b274a384f3eb5b5fe7d781a Mon Sep 17 00:00:00 2001 From: tomit4 Date: Mon, 27 Jul 2026 18:06:45 -0700 Subject: [PATCH] :construction: Setup for 7.2 --- chapter_7/exercises.md | 402 +++++++++++++++++++++++++++++++++++++ chapter_7/notes.md | 152 ++++++++++++++ chapter_7/test_yourself.md | 41 ++++ 3 files changed, 595 insertions(+) diff --git a/chapter_7/exercises.md b/chapter_7/exercises.md index 7a8be90..3433269 100644 --- a/chapter_7/exercises.md +++ b/chapter_7/exercises.md @@ -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. diff --git a/chapter_7/notes.md b/chapter_7/notes.md index bae55fe..f6b2ebf 100644 --- a/chapter_7/notes.md +++ b/chapter_7/notes.md @@ -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.]_ diff --git a/chapter_7/test_yourself.md b/chapter_7/test_yourself.md index 640a492..e4f633e 100644 --- a/chapter_7/test_yourself.md +++ b/chapter_7/test_yourself.md @@ -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 _____.