The textbook is The Tools of Mathematical Reasoning by Tamara J. Lakins. The author has posted some errata. The following comments augment that list.

page 3, paragraph preceding Table 1.2

The statement that “or” is inclusive needs a disclaimer that the words “true or false” are not equivalent to the symbols “T \(\vee\) F”. Throughout the book, the words “true or false” convey an exclusive “or”, as in the first sentence of Example 1.1.4 on page 4.

page 40, line 8

To justify the statement that “every integer \(k\) either satisfies \(k\le 33\) or \(k\ge 34\)” requires some knowledge beyond the Basic Properties of Integers 1.2.3 on page 23. One needs to know an additional property, for instance, that \(1\) is the smallest positive integer. A similar comment applies to Exercises 2, 3, and 4 on page 43.

page 40, last paragraph, first line

For “the sum and product of two rational numbers is rational” read “the sum and the product of two rational numbers are rational”. (The plural subject, “sum and product”, takes a plural verb, “are”.)

page 49, Definition 2.4.2, fourth line

Insert a comma after \(\lim\limits_{x\to a} f(x)=L\).

page 73, Notation 4.1.10

The list of notation should include a definition of \( (-\infty,\infty)\), since this notation is used later on (for instance in Exercise 24 on page 82 and in Exercise 11 on pages 201–202). As you probably learned in your calculus course, the interval notation \( (-\infty,\infty)\) means \(\mathbb{R}\), the set of all real numbers.

page 84, line 15

The word “Hence” is wrong, for what is being stated is not a deduction from the preceding steps but rather an additional input to the proof (namely, an invocation of the remark on the preceding page that finite unions can be written unambiguously without parentheses). The word “Moreover” would be an appropriate substitute.

page 88, Exercises 1–5

Rigorous solutions of these exercises are not possible using only the Basic Properties of Real Numbers 2.1.4. You may assume the Archimedean Property from the later Section 9.3. In other words, you may assume that for every real number \(x\), there exists an integer greater than \(x\). The notation \(\lceil x\rceil\) (the ceiling function) denotes the least integer greater than or equal to \(x\).

In Exercise 4, replace the assumption that \(i\ge 1\) with the assumption that \(i\ge 2\). The issue is that \(A_1\) would be the undefined expression \( [0,0]\). On page 73, the author has defined the interval \([a,b]\) only when \(a\) is strictly less than \(b\).

page 104, Exercise 12

The referenced Table 6.2 can be found on page 152 in Section 6.5.

page 104, Definition 5.2.1

Almost everybody agrees on the meaning of the symbols \(g\circ f\), but there is no agreement on how to put the concept into words. The author says, “the composition of \(f\) and \(g\),” but many other writers say, “the composition of \(g\) and \(f\)” with the names of the functions in the opposite order. Also common is “composition … with” instead of “composition … and.” The word “composition” is more common than “composite” as a noun, but the phrase “composite function” is unobjectionable.

page 106, line 8

For “\(g(x-1)\)” read “\(g(x-2)\)”.

page 114, Exercise 10

In the definition of \(C(x,y)\), the scope of the sum is potentially ambiguous. The intent is for \(y\) to be outside the sum. Clearer would be \(y+ \displaystyle \sum_{i=1}^{x+y-1} (i-1) \).

page 122, second Warning

The statement that “the inverse image \(f^{-1}[B]\) under \(f\) has nothing to do with inverse functions” is an exaggeration. If the function \(f\) happens to be invertible, then the two different meanings of the notation \(f^{-1}[B]\) actually coincide.

page 133, line 17

For \( a- (b+1)q\) read \( a-(q+1)b\).

page 140, Exercise 4

The exercise depends on Exercise 3, which shows that \(d\) is a linear combination of \(a\) and \(b\).

page 143, Base Case

Invoking Corollary 6.3.6 is overkill. The definition of prime number implies that \(p_1\) cannot be written as the product of two (or more) integers larger than \(1\), so it is immediate that \(s=1\).

page 162, Exercise 10

The statement is confusing because the relation is called both \(\sim\) and \(R\). Replacing \(R\) with \(\sim\) would make the notation self-consistent.

page 170, four lines above Theorem 8.2.1

\(|Y|=m\) should be \(|X|=m\).

page 171, statement of Theorem 8.2.5

Change “let \(n\), \(m\ge 0\) be natural numbers” to “let \(n\) and \(m\) be nonnegative integers” or simply “let \(n\), \(m\ge 0\)”. The issue is that \(0\) is not a natural number (see the definition in Table 1.10 on page 10).

page 173, first line

Delete “\(m\), \(n\in \mathbb{N}\) and”. We may not fix \(m\) and \(n\) at this point in the discussion, for \(m\) and \(n\) have already been prescribed explicitly in the preceding paragraph (as well as implicitly in the statement of the theorem).

page 178, Exercise 9

Append a period at the end of the first line.

page 187, Exercise 18

Append a period to the word “uncountable” (which is the end of the second sentence).

page 196, Proposition 9.2.8

The word “Conversely” is misleading, for the subsequent statement is not the converse of the preceding statement.

page 201, Exercise 8, first line

For \( \{cx\in x\in S\}\) read \(\{cx \mid x\in S\}\).