#### 6

- Page 6 of the preface contains a grammatical in the last paragraph, 4 lines down. There is a oddly placed "the" near the end. I believe it should read either, "By reading the lesson, then you will acquire...." or "By reading the lesson, you will acquire...."
- Example 1.5 "Consider the function f(x) = x^2 as a function from the integers (Z = {-3, -2, -1, 0, 1, 2, 3}) to the integers." There should be ellipses before -3 and after 3 to show that the function can take on more values. With the correction it should read: (Z = {...-3, -2, -1, 0, 1, 2, 3...})

#### 16

- Towards the lower middle of the page that begins #(O)=98+97+96...2+1 shouldn't it be #(O)=99+98+97+96...2+1? The 99 was left out
- In the first paragraph it repeats the word can twice after the comma..... "....., can can be uniquely named but the set..."
- When A is split into A1 and A2. The notation for A2 is incorrect because j should not be included since handshake is only between Alice and the initiator. It should read A2= {(pi, a)}:1 <=i <=99}.
- Towards the lower end of the page where A is split into A1 and A2. A1={(a, pi):1<= i, j <=99}. The j shouldn't be part of the notation because the handshake is only between Alice and the initiator.

#### 26

- #(I) and #(II) do not correspond with #(F) and #(S). Either change both to I and II or both to F and S.
- The statement " $\forall$ a, A [not [a in A] or [a not in A] ]" is incorrect. The correct statement should be " $\forall$ a, A [ [a in A] or [a not in A] ]" because "not [a in A]" is the same as "[a not in A]." -Paula Lin

#### 27

- For Exercise 3.7, it says "Your job is to paint bookcases one of three different colors or to leave the bookcase unpainted. You have eight different bookcases and all of the book cases have a different height. In how many ways can you do your job?" The first error (a barely noticeable one) is the space between "book" and "cases." The second is not really an error, but rather something I found to be superfluous. Because the question doesn't really concern the height of the bookcases, mentioning that the bookcases have a different height isn't really necessary. I believe the issue of height was more important in Exercise 3.9.
- On 3.6 there is an added "s" at the end of drug that should not be there. There is one at the being of the question and it reads, " three drugs types are to be tested . ." and again towards the end of the question which reads, " it is not necessary that all drugs types are used."

#### 36

- 4.9 should read "How many ways can a frog jump..." rather than "How many ways can a from jump..."
- There is a minor typo in 4.4 where it states "in how many way can they be returned" instead of "in how many ways can they be returned"
- On Exercise 4.12, the book states "If the T.A. and professor cannot stand next to each other, how many distinct pictures can be taken." The period should probably be a ?
- In the trial edition. Top of 36, for the proof of the proposition 4.1* on page 35. While proving subproposition 2, the first line should read: "Suppose that $y \in f(A_1 \cap A_2 )$. Then $y = f(x)$ for some $x \in (A_1 \cap A_2)$ ... " not " Then $y = f(x)$ for some $x \in f(A_1 \cap A_2)$ ... " *Also, the proposition numbering is inconsistent with the rest of the section.
- On exercise 4.9, the first sentence begins, "How many ways can a 'from' jump on seven lily pads..." when in reality the sentence should read "How many ways can a 'FROG' jump on seven lily pads..."

#### 43

#### 59

- In the proof of Proposition 8.1: 2nd paragraph, 2nd line: Correction: "Therefore, $mn = 2(2kl)$ and so $mn$ is even." The current print says "$mn = 2(2ml)$", which is incorrect.
- In the third sentence, it says the Aztec and Mayan peopleS, when i think it should read the Aztec and Mayan people, later in the sentence it also says, "...and as does Dzongkha..." which makes no sense, I think the and isn't supposed to be there

#### 71

- On page 71, near the top where there is an equation of #(O)= (n m1)x (n m2)... The third one on the book wrote (n m2) is it supposed to be (n m3).... (n mk) instead?
- In the bottom part of Figure 7., there is an incorrect statement made. A point on the line with -A is identified as "-a < -sup(A)" however it should actually read "-a > -sup(A)."

#### 74

- Under example 8.7 it said that we will follow the second approach that we used in Example 8.5. Should it be Example 8.6 instead because you already wrote that the first approach used in Example 8.5 in the previous sentence?
- In the proof of Proposition 11.2, 2nd paragraph, in the third and penultimate lines from the end. The $p$ mentioned are missing its subscripts. Corrected: "Therefore, $p_1$ divides $b$. Therefore, $p_1$ is a common factor of both $a$ and $b$."

#### 99

- In exercise 11.9, the word "envelop" should be spelled "envelope."
- Question 11.9 should read, "There are n letters and n envelopes." Not, "There are n letters and n envelops." Dylan
- In exercise 11.3, in the sentence "Otherwise, you add 1 to the value of the of the side showing on the die," the fragment "the of" is repetitive.

#### 106

- The sentence beginning with "Bag 1 contain 1 red balls and 89 green ball" in example 12.3 should be "Bag 1 contains 1 red ball and 89 green balls."
- The Fibonacci sequence in Example 16.3 is defined incorrectly. It should read: $a(1) = 1$, $a(2) = 1$, and $a(n+2) = a(n) + a(n+1)$. Having $a(2) = 2$ as it currently reads does not match the sequence listed in the premise: $\{ 1, 1, 2, 3, 5, 8, ...\}$.

#### 107

- In exercise 12.2 it said that you toss the coin there times. Is it supposed to be three times?
- In the proof of Proposition 16.1, after the function definition of $g(z)$. The function $h(n) = g(b(n))$ is a bijection from $\mathbb{N}_{0} \rightarrow \mathbb{Q}$. The function $b$ is defined on the bottom of page 106 as $b:\mathbb{N}_{0} \rightarrow \mathbb{Z}$, so this should carry over to the definition of $h: \mathbb{N}_{0} \rightarrow \mathbb{Q}$.

#### 116

- In exercise 13.13, it should be "the first is a boy" instead of "the first is boy."
- In exercise 13.14, it should be "that one is a boy" instead of "that one is boy."
- In exercise 13.11 it should be "the first is a boy" not "the first is boy."
- In exercise 13.12, it should be "one is a boy" instead of "one is boy."

#### 138

- Thm 20.2, first paragraph of the proof, 3rd line from the end. The line beginning "if $\alpha > a $ should read "if $\alpha >b$...". Then does the rest of the proof make sense.
- In the proof for Thm 20.2, paragraph 2. A set $A$ is mentioned without definition. The set $I$ should be the not sequentially compact set in mentioned.

#### 141

- In example 21.1, we know that $N > \frac{22}{\epsilon}$ and $n,m>N$. Then later in the proof, we only conclude that $\frac{11}{m} + \frac{11}{n} \geq \frac{\epsilon}{2} +\frac{\epsilon}{2}$ when we could conclude right there that it's strictly greater (and that's the only place where we can do that so it needs to be done there).
- 16.1 the last sentence has an unecessary "that" towards the end.
- In the definition of the Cauchy Sequence, it should read: "a sequence $a_n \in \mathbb{F}$ is a Cauchy sequence, if, given an $\epsilon >0$, $\exists N\in\mathbb{N}$ such that $m,n > N$ implies $|a_n - a_m| < \epsilon$. Without the $m$ index, this definition does not make any sense.

#### 146

- In 22.2, \sum\limits_{n=5}^infty ($\frac{x}{y}$)^n should equal 1/16 and not 1/8
- Indexing of summation presented in the hypothesis is incorrect. \[\] It should read: "Suppose that $k \in \mathbb{R}$ and $\sum_{n = 1}^{\infty} a_n$ is convergent." $k$, as an index, as not been used in the scope of the summation.

#### 147

- Index of the last summation should read: \[\] $\sum_{n = 1}^{\infty} \frac{1}{n} = ...$. The current index of $k = 1$ is incorrect, as there is no correspondence to $k$ present in the summation.
- The last line of the proof to show that sum of $\frac{1}{n^2}$ is convergent, mistakenly says " s_{N}$ =$\frac{1}{1}$ - $\frac{1}{N}$ < 1." but should actually read " =$\frac{2}{1}$ - $\frac{1}{N}$ <2." Because in the line above, $\frac{1}{1}$ + $\frac{1}{1}$ remain.
- Example 22.4. At the last of the proof, 1/1-1/N<1 should be 2/1-1/N<2 -Sun Kim

#### 152

- In the bottom half of the proof at the top of the page, the book mistakenly says: Therefore, if n>m+1, either s_{m}$ < s_{n}$ < s_{m+1}$ or s_{m}$ < s_{n}$ < s_{m+1}$ . But this is repeating the same inequality. The second part should have the inequalities flipped and read s_{m}$ < s_{n}$ < s_{m+1}$ or s_{m}$ > s_{n}$ > s_{m+1}$ .
- Question 23.1 mistakenly repeats the inequality, but should ask: either s_{m}$ < s_{n}$ < s_{m+1}$ or s_{m+1}$ < s_{n}$ < s_{m}$ .

#### 158

- Example 24.2 at the bottom of the page incorrectly sets | $\frac{ a_{n+1}$ }{ a_{n}$ }$ | = | (1/2)($\frac{1}{n+1}$ | but the correct answer is | $\frac{ a_{n+1}$ }{ a_{n}$ }$ | = | (2)($\frac{1}{n+1}$ | .
- Example 24.2 claims, in the concluding sentence, that the root test shows that the sequence in the example converges. We did not use the root test here, but the ratio test.

#### 170

#### 189

- For the last paragraph of page 189 for 131 A, in line 7, "this would mean that f is not one-one. Since f is assumed to be one-one, it is not possible that...". Formally, it should be "this would mean that f is not one-to-one. Since f is assumed to be one-to-one, it is not possible that..." for the word "one-to-one". I was a little confused at the first time I read that paragraph, though it makes sense.
- The last paragraph of page 189 for 131 A, from line 3, "if if f is neither strictly increasing not strictly decreasing but one-to-one, there....", it should be "if if f is neither strictly increasing not strictly decreasing by one-to-one, there....", using "by" instead of "but".