# Mathematical Thinking - Changes to 2nd edition

These items are listed in several categories. Category 1 lists items discovered since the third printing in 2001. Category 2 lists mathematical and notational corrections that were implemented for the second and third printings in 2000 and 2001. Category 3 lists minor changes such as the fixing of typographical errors, also implemented for those printings. Readers who have the third printing (or later) of the second edition need only check Category 1. (Contributors are named in parentheses.)

Note: Some corrected pages for the second printing were processed with the wrong command. This has introduced some cross-referencing errors and perhaps other errors. The following have been discovered.

• p47 - Exercise 2.31: the reference should be to Remark 2.34 as in the first printing, not Remark 2.88 (Greg Marks)
• p74 - Exercise 3.53: the reference should be to Theorem 3.24 as in the first printing (or Lemma 3.23), not Remark 3.89 (Jim Faran)
• p74 - Exercise 3.54: the reference should be to Lemma 3.23 as in the first printing, not Lemma 3.88
• p75 - Exercise 3.61: the reference should be to Problem 3.4 as in the first printing, not Problem 3.69 (Dan Grayson)

## Category 1: Corrections and Clarifications since the 2001 printing

### . . . not yet implemented in the text

• p44 - Exercise 2.11: students may misinterpret the dollar as being a bill rather than a coin, so "dollar" should be changed to "quarter" (Paul Hagelstein)
• p46 - Exercise 2.17: the statement should be "Consider f:\RR->\RR such that f(x)\ne1 for all x. Let g(x)=x/2+x/(f(x)-1). Prove that if x\ne0 and g(x)=g(-x), then f(x)f(-x)=1. (Jeremy Weissmann)
• p48 - Exercise 2.37: statement C should also be in quotation marks: "x^2=1" (Laszlo Liptak)
• p57 - Proposition 3.19: the limits of the last summation in the proof should be from i=1 to i=k+1, not i=k (Blair Goodlin)
• p59 - after solution 3.22: the reference should be to Chapter 5, not Chapter 9
• p67 - Example 3.35: at the end of the displayed equation, the term 2k2 should be 3k2. After the display, "we need only verify that k(3k+1)\ge 15. Unfortunately, this requires k\ge3" (Greg Marks)
• p88 - Remark 4.40: there is no set of all finite sets; this notion runs afoul of Russell's Paradox. Fortunately, it is not necessary to think of size in these terms. The first paragraph of 4.40 should be deleted, replaced by the remark that listing the elements a1,...an in a set A of size n is equivalent to explicitly defining a bijection from [n] to A (Dan Grayson)
• p95 - Exercise 4.10: the domain and target for both f and g should be specified as the set of real numbers (Ricardo Velasco)
• p97 - Exercise 4.28: the problem should state the polynomial as f(z)=az^3+bz^2+cz+d and make clear that the request is for necessary and sufficient conditions on the coefficients a,b,c,d so that f is an injective function. Rather than "geometric arguments", the hint should suggest using a substitution of the form z = sx+t to reduce the problem (with proof) to the special case h(x)=x^3+rx
• p98 - Exercise 4.46c: a prohibition of using the Schroeder-Bernstein Theorem will be added, since that would yield |A|=|B| immediately (Jerry Folland)
• p114 - Example 5.40: Each two-line form here should have 6 before 7 in the lower line (Bryan Bowman)
• p121 - Exercise 5.53: (discovered in 2022) "n-1 permutations" should be "n-1 transpositions", and "n/2" should be "\floor{n/2}" in the last part
• p122 - Exercise 5.64: the upper limit of the summation in the hint should be k+1, not k (John Palmieri)
• p135 - Exercise 6.21: "x\in\RR" should be "x\in\ZZ" (Kevin Ford)
• p136 - Exercise 6.38: second part should read "Conclude that if 2n+1 is prime, then n is a power of 2"
• p136 - Exercise 6.39: There are now 39 known Mersenne primes; the website www.mersenne.org gives more references than the specific article mentioned in the second edition (Dan Grayson)
• p138 - Exercise 6.54: The hint is somewhat confusing. A better hint is: "Assume that x< y< z. Use the binary expansion of an appropriate number to show that if (x,y,z) is achievable, then (x',y',z') is achievable, where y'=y-\floor{y/x}".
• p152 - Exercise 7.21: The emphasis on modular arithmetic is misleading. What is intended is an inductive proof.
• p157 - Definition 8.6: "no common factors" should be "no common factor greater than 1" (Benjamin Allgeier)
• p165 - after Theorem 8.23: the paragraph should refer to the special case in Exercise 3.41. The word "requires" should be "follows from"; there are examples of discontinuous functions for which the statement fails, but it holds for more than just the continuous functions
• p167 - Exercise 8.24: The given hint is somewhat mysterious. The problem can be solved by mimicking the proof of Theorem 8.14.
• p174 - Example 9.12: "inside an inscribed equilateral triangle" should be "inside some inscribed equilateral triangle"
• p180 - Definition 9.28: "defined on \NN by" should be "taking values in \NN, defined by" (Jerry Folland)
• p198 - Exercise 10.10: the hint in the back that each 100-yard field contains its endpoints should be part of the problem statement (Jerry Folland)
• p215 - Theorem 11.40: "trees with n-1 edges have n-2 vertices" should be "trees with n-1 vertices have n-2 edges" (Crystal Hwang)
• p241 - Definition 12.24: in the notation for the characteristic polynomial, the summand should be cixn-i, not cian-i (Charles de Matas)
• p252 - Exercise 12.28: If differentiation (of the characteristic polynomial) is not allowed, then the generating function method (and partial fraction expansion) must be used to solve this.
• p256 - before Definition 13.3: to the fifth decimal place, the expansion of \sqrt 2 is 1.41421, not 1.41423 (Asha Khazad)
• p289 - Exercise 14.31: the hint should end with "unless x1=\sqrt 2)" (Tyler Smith)
• p310 - Lemma 16.15b: the statement should be "If s(h) -> 0, then e(s(h))/s(h) -> 0 as h -> 0." The proof should be changed accordingly (Christian Stapfer)
• p311 - Theorem 16.16(d): The right side of the last equation in the proof should be "-g'(x)/[g(x)]²", not "g'(x)/[g(x)]²" (Willard Ramsey)
• p327 - Example 16.44: the second reference to Exercise 14.44 should be to Exercise 14.63 (John Hines)
• p333 - Exercise 16.37: in part (b), "(Af)" should be "Af"; in the hint, "hx" should be "hx"
• p335 - Exercise 16.66: the upper index "n" should be "\infty"
• p336 - Exercise 16.67: the upper index "n" should be "\infty"
• p358 - Exercise 17.17: "bijection" should be "continuous bijection"
• p380 - Definition A.29: at the end in the definition of -\alpha there should be a colon: -alpha={ < -a >: a\in\alpha} (Laszlo Liptak)
• p391 - Hint 9.39: "ballet lists" should be "ballot lists"
• p398 - Hint 18.12: this is the hint for Exercise 18.15

## Category 2: Mathematical Corrections made in early printings

### . . . (2a): submitted for the third printing (April, 2001)

• p22 - Exercise 1.20: the end of the first sentence should be ", where a not 0." (Carl Jockusch)
• p55 - Corollary 3.14: n should be restricted to positive integers (Rick Laugesen)
• p72 - Exercise 3.10: each instance of "2n+1" should be "2n-1" so that the first instance treats a set of size one (Carl Jockusch)
• p74 - Exercise 3.54: "F" should be "f"
• p75 - Exercise 3.57: "n\ge2" should be "n\ge3"
• p77,78 - before Example 4.4 and Theorem 4.6: the references to Theorem 4.7 should be to Theorem 4.6 (Carl Jockusch)
• p84 - Proposition 4.25: "f(n) > f(y)" should be "f(x) < f(y)" (Dennis Turpin)
• p85 - before Definition 4.28: "is contained in" should be "is" (Dennis Turpin)
• p89 - after Definition 4.43: concerning Z, "we do consider" should be "we do not consider" (Dennis Turpin)
• p94 - last paragraph: the reference to Exercise 44 should be to Exercise 49
• p101 - Definition 5.6: add "nonempty" before "subsets"
• p112 - before Theorem 5.36: after "we also have" should come "fn=" (Carl Jockusch)
• p121 - Exercise 5.53 "at most n-1 permutations" should be "at most n-1 transpositions". Also, the correct lower bound in the last line is the floor of n/2
• p127 - Algorithm 6.15: with the convention that gcd(0,0)=0, there is no need to exclude the pair (0,0) (Rick Laugesen)
• p135 - Exercise 6.16: add the requirement "with b not zero" (Carl Jockusch)
• p135 - Exercise 6.21: the second part needs the restriction that x is an integer
• p136-7 - Exercise 6.43: the algorithm as stated does not work for all pairs of nonnegative integers. The simplest correction is to require that a and b are not both even, as in the first edition (Rick Laugesen). A more thorough solution is to change option (2) to apply only when both are nonzero and one is even; when both are nonzero and even, divide both by 2 and add 1 to a running count of common factors of 2.
• p138 - Exercise 6.60: the reference to "exercise 50" should be "Exercise 59"
• p140 - Definition 7.9: the word "distinct" should be omitted (Carl Jockusch)
• p152 - Exercise 7.21: this problem should request a proof by induction and properties of divisibility
• p153 - Exercise 7.29: "Test for divisibility by n" should be "Test for divisibility by s"
• p164 - In the remark about Fermat's Last Theorem, "integers" should be "positive integers" (Carl Jockusch)
• p184 - Exercise 9.4: The question should ask about the probability of the intersection (Rick Laugesen)
• p186 - Exercise 9.16: The problem should be restricted to n at least 4 to avoid special cases
• p197 - after solution 10.14: (typesetting error) the notation for the intersection of the sets Ai over i in S should be the same in the third line as in the first line (Carl Jockusch)
• p199 - Exercise 10.21: "132 days" should be "132 games"
• p228 - Exercise 11.2: To avoid annoying special cases, it would be better to restrict $n$ to exceed 12
• p258 - Solution 13.7: In the display, the squares should be taken before the min or max (Rick Laugesen)
• p264 - second paragraph: "Chapter 5" should be "Chapter 4" (Rick Laugesen)
• p265 - Theorem 13.25: in the last line, "Ln" should be "ln"
• p266 - Theorem 13.27: Since we are listing canonical expansions, we should change [0,1] to [0,1). In the figure, there should be a decimal point to start the expansions and an ellipsis at the right end of each (Rick Laugesen)
• p270 - Exercise 13.31 missing parenthesis: "(1+(1/n)n" should be "(1+(1/n))n"
• p275 - Proposition 14.11: "Choose N2>N1\epsilon/(2M) should be "Choose N2>N1M/(2\epsilon)"
• p291 - Exercise 14.57: references should be to Example 12.36 and Solution 12.23
• p302 - Theorem 15.24: add "there" at the end of the theorem statement
• p304 - Exercise 15.1: "f(x)" should be "f(k)" (Carl Jockusch)
• p358 - Exercise 17.18: in the second part, a and b should be reciprocals of positive integers (also final period missing)
• p394 - Hint 13.16: The suggested method uses the geometric series, which is not discussed until Chapter 14. One can instead multiply by 26 to obtain an equation for the desired value
• p397 - Hint 17.22: the specified endpoints should be t=1 and t=1+1/x, not t=x and t=x+1
• p397 - Hint 17.25: The breakpoints should be bk/n, not xk/n

### . . . (2b): made for the second printing (February, 2000)

• p17 - Proposition 1.45: In F4, "<" should be "<="
• p24 - Exercise 1.50: the problem should be stated for intersections, not unions; with unions part (b) is wrong (Margit Messmer)
• p47 - Exercise 2.33: third sentence should be clarified as "The third child sees the hats on the first two, the second child sees the hat on the first, and the first child sees no hats" (Josh Kim)
• p80 - Example 4.9: with the given formula, the function is a bijection from N\cup 0 to Z. To obtain a function from N to Z, f(n) should equal -(n-1)/2 when n is odd, which requires slight modifications in the later comments (Margit Messmer)
• p120 - Exercise 5.40: the sum should extend to n, not n-1 (Karen Mortensen)
• p188 - Exercise 9.38: in the example given in part (b), the first row should be 1247, not 1246
• p202 - Problem 11.1: the names of the land masses Y and Z should be switched
• p268 - Exercise 13.12 "(xn+yn)" should be "(xn+yn)/2"

## Category 3: Typos and Clarifications corrected in early printings

### . . . (3a): submitted for the third printing (April, 2001)

• p12 - Definition 1.31: in last line, "|f(x|" should be "|f(x)|" (Carl Jockusch)
• p15 - Application 1.38: "Babylonian problem of Chapter 1" - this is Chapter 1, so those three words are superfluous
• p28 - "grammtically" should be "grammatically" (Carl Jockusch)
• p47 - Exercise 2.29: "on" should be "defined on"
• p48 - Exercise 2.39: "if and only if" should be "if and only if the following two conditions hold:"
• p51 - Theorem 3.6 (start of proof): there is a missing space before "is true" (Carl Jockusch)
• p59 - Lemma 3.23: for the case a=0, the notation assumes that 00=1 (Carl Jockusch)
• p73 - Exercise 3.37: the restriction on n should be to natural numbers larger than 1 (Carl Jockusch) (placed in hint at the back due to lack of space)
• p74 - Exercise 3.53: the first "f(x)" should be "f"
• p74 - Exercise 3.54: "c" should be "cn" (better is to refer to the coefficients generally rather than by formula)
• p96 - Exercise 4.22: the hint should begin with "Compare with Solution 4.46" (Rick Laugesen)
• p97 - Exercise 4.33: part (a) needs a period
• p115 - last paragraph: eliminate extra "depends" (Dennis Turpin)
• p118 - Exercise 5.5: "pairs" should be "one pair"
• p120 - Exercise 5.34: "integers sizes" should be "positive integer sizes"
• p128 - Example 6.19: Eliminate "reduced" from the first paragraph and replace "the reduced equation" with "2x+5y=1" in the second paragraph (Carl Jockusch)
• p135 - Exercise 6.23: the set braces should be dropped (Carl Jockusch)
• p141 - Example 7.14: in the last line, "be" should be "by" (Carl Jockusch)
• p211 - Definition 11.28: "place" should be "placed"
• p250 - Exercises 12.1-12.5: these should be marked "(-)"
• p279 - Definition 14.20: the "th" in the definition of partial sum should be in roman
• p384 - Hint 1.8: "need not" should be "do not"
• p387 - Hint 4.49: the union symbol should replace the two commas in the second line (Carl Jockusch)

### . . . (3b): submitted for the second printing (February, 2000)

• p120 - Exercise 5.25: this should be reworded slightly since we have not defined the factorial function at negative integers (Margit Messmer)
• p121 - Exercise 5.56: for second part, drop the unnecessary -1 and add the suggestion to form permutations from subsets
• p135 - Exercise 6.30: the statement should end with "when n is a nonnegative integer"
• p167 - Exercises 8.21,24: change p to f for consistency of notation
• p185 - Exercise 9.14: after "always ahead of B", add "(after the beginning)"
• p187 - Exercise 9.29: "into pairs up at random" should be "into pairs at random"
• p201 - Exercise 10.35: reworded to clarify
• p268 - Exercise 13.3: add "other than `not'"
• p288 - Exercise 14.11: the second part (a) should be part (b)
• p326 - Lemma 16.61: there is an end-of-proof mark where the proof is only halfway finished
• p327 - Theorem 16.65: when we introduce fn(x) in the proof, we can choose any single n\ge N. Just before the display should be "any fixed n\ge N"
• p332 - Exercise 16.25: at the end of the problem, add "(Assume the production cost is zero)"
• p392 - Hint 10.19: change to "Show that with 989 keys, some 90 members can't open all the rooms."