Section 1.7

• 14. First note that k^3 - (k-1)^3 = 3k^2 - 3k + 1. Then sum both sides as k = 1 to n. The left side is just n^3 by telescoping. On the right we have three terms, the last two of which we know and the first one we solve for. After some algebraic manipulation, the answer is n(n+1)(2n+1)/6.
• 16. n! = PROD(i=1,n) i
• 18. 288
• 20. a) 1, -1, 2 -2, 4, -4, 5, -5, ...; b) 5, -5, 10, -10, 15, -15, 20, -20, 25, -25, 30, -30, 40, -40, ...; c) countable. Set up a 2-dimensional matrix with values with one additional 1 to the right of the decimal point for every column you move to the right, and one additional 1 to the left of the decimal point for every row you go down. Can enumerate these by stepping through diagonals that start in an element in the top row and proceed down and to the left. d) not countable. Can prove using the diagnolization argument used to prove that the set of reals is uncountable.
• 22. subset of a countable set is countable. Removing all elements of the enumerated sequence that are not in the subset still leaves a sequence that can be enumerated.
• 24. Enumerate the union of two countable sets by interspersing their elements, like a1, b1, a2, b2, ...
• 26. Enumerate them as 1/q, 2/q, ..., (q-1)/q. This is countable for a single q. Taking the union of these for all q = 1, 2, ... we are taking the union of countable sets, which is countable (exercise 25). Imagine arranging these fractions in a 2-dimensional array as we did for exercise 20d.
• 28. This amounts to proving that the number of triples (a,b,c) with a, b, and c being integers, is countable. There are a countable number of pairs (b,c), because for a given b, there are countably many c's. Since there are a countably many number of b's, we have the union of countable sets, making another countable set. Same argument extends to triples.
• 30. We know from Example 12 that the set of real numbers between 0 and 1 is uncountable. Let us associate to each real number in this range (including 0 but excluding 1) a function from the set of positive integers to the set {0,1,...,9} as follows: If x is a real number whose decimal represenation is 0.d1d2d3d4... (with ambiguity resolved by forbidding the decimal to end with an infinite string of 9's), then we associate to x the function whose rule is given by f(n) = dn. Clearly this is a one-to-one function from the set of real numbers between 0 and 1 and a subset of the set of all functions from the set of positive integers to the set {0,1,...,9}. Two different real numbers must have different decimal representations, so the corresponding functions are different. (A few functions are left out, because of forbidding representations such as 0.2399999...) Since the set of real numbers between 0 and 1 is uncountable, the subset of functions we have associated with them must be uncountable. But the set of all such functions has at least this cardinality, so it, too, must be uncountable (by Exercise 23).