"Every natural number has a unique immediate successor." Why is the word "unique" important for the discussion that follows?
"We will write it as 1." Could we have used another notation? Why or why not?
Page 4
"(n + 1) - 1 = n." What does this statement mean in words?
"The predecessor of the successor of a natural number n is just the number n." Why do they use the article "the" and not "a" in this sentence? Is this a stylistic choice, or is there a reason for such a choice?
"The predecessor of the successor of a natural number n is just the number n.” How would this statement read with different notation for "predecessor" and "successor?"
"If n < m and m < k, then n < k." Explain how the discussion that follows may be viewed as a proof of this statement?
Page 5
"(1 + 1) + 1 = 3." Rephrase this statement in words.
Page 6
"It is clear that the definition of addition is just the rearrangement of the parenthesis around 1s and +s." Elaborate on this statement.
Page 7
"If m and n are natural numbers, then n < n + m." Explain how this follows from the associative property of addition.
"If k, m, and n are natural numbers and n < m, then n + k < m + k." Explain why you need the associative property of addition to justify this statement.
Page 8
"Multiplication of natural numbers is commutative." Why do they use the same word, "commutative" as they did for addition? What does "commutative" actually mean?
Page 9
"Multiplication of natural numbers is associative." Why do they use the same word, "associative" as they did for addition? What does "associative" actually mean?
"n ⋅ 1 = 1 ⋅ n = n." Explain how this follows from the definition of multiplication.
Equations (1.14) and (1.15). Explain what these equations are saying.
Section 1.2
Page 13
Equation (1.26). Explain why we need to write n ⋅ 1 + n ⋅ 1 = n ⋅ (1 + 1) before writing 2 n.
Page 15
"Because every natural number is an eventual successor of 1, we will eventually know that the statement is true for any number." Explain why a proof by induction amounts to show that a statement is true for all natural numbers.