1.6 The Positive Integers and the Induction Principle 27
PROOF. Let K be a nonempty subset of N. If K contains 1, then K has a least element, namely 1, and so there is nothing to prove.
Suppose then that K does not contain 1. Let S be the set of all positive in-tegers n for which it is true that K contains none of the integers 1,2, ... , n. (For example, if K were the set {10, 20, 30, S would be the set {1, 2, 3, , 9}. Fif-teen would not belong to S, because 10 < 15, and 10 belongs to K.)
We know that (1) S contains 1, because K does not contain 1. If it were true that (2) S is closed under the operation of adding 1, then S would contain all of the positive integers, and K would be empty. Therefore S must not be closed under the operation of adding 1. Hence there is an integer n such that n be-longs to S and n + 1 does not. This means that K contains none of the numbers 1,2, ... , n, but does contain n + 1. It follows that n + 1 is the smallest element of K. ❑
(In the example given above, the smallest element of K is obviously 10. You should check to see that 10 is the number that we get when we apply the gen-eral proof to this particular set K.)
The choice between Theorems 1 and 2 is merely a matter of taste. But there are times when the well-ordering principle is easier to use than either of them. (See, for example, the chapter on the theory of numbers.)
In the following problem set (and hereafter), when you prove things by in-duction, you are supposed to use the induction principle in one of the forms given in this section. Thus your proofs should be in one of the following forms.
"Let S be .... Then (1) 1 belongs to S. Proof.
(2) For each n, if n belongs to S, then n + 1 belongs to S. Proof: ... . By the induction principle, the theorem follows."
Or:
"For each n, let P,, be the proposition that .... Then:
(1) P„ is true. Proof.
(2) For each n, Pn implies Pn+1. Proof: ... .
Therefore, all the propositions Pn are true, which was to be proved."
If you use either of these forms, the reader can tell what principle you are applying. Very often, "proofs by induction" are written in forms that give the reader no inkling of what the induction principle is supposed to be.
28 The Algebra of the Real Numbers
2. Show that the sum of the first n odd positive integers is n2. That is, 1 + 3 + 5 + • • • + (2n — 1) = n 2.
3. Show that for every n > 0,
2 13 + + • • • +
2(n+ + 1)) .
4. Assume that for every positive real number x there is a positive integer n > x. (This is true; it is a consequence of the Archimedean property of the real numbers, discussed in Section 1.8.) Show that for every positive real number x there is a nonnegative integer n such that n x < n + 1.
5. Now show that for every real number x there is an integer n such that n x <
n+ 1.
*6. The game known as the Towers of Hanoi is played as follows. We have three spin-dles A, B, and C, of the sort used as targets in quoits. On spindle A we have a stack of n disks, diminishing in size from bottom to top. These are numbered 1, 2, ... , n, in the order from top to bottom. A legal move in the game consists in taking the topmost disk from one spindle and placing it at the top of the stack on another spindle, provided that we do not, at any stage, place a disk above a disk which is smaller. (Thus, at the outset, there are exactly two legal moves: disk 1 can be moved either to spindle B or to spindle C.) The object of the game is to move all the disks to spindle B. Show that for every positive integer n, the game can be completed.
*7. Let N. be the number of moves required to complete the game with n disks. Show that for each n,
pn, = 2p„ + 1 .
8. Given that p, = 1, and that p„+1 = 2p„ + 1 for each n, show that pn = 2" — 1.
(Since 210 = 1024, this means that the game with 20 disks requires more than a mil-lion moves.)
1.7 The Integers and the Rational Numbers
If we add to the set N the number 0 and then all of the negatives of the num- bers in N, we get all of the integers. The set of integers is denoted by Z. Thus
Z = {..., —3, —2,
If a number x can be expressed in the form of
p/q,
where p and q are in-tegers and q 0, then x is called a rational number. The set of all rational num-1.7 The Integers and the Rational Numbers 29
bers is denoted by 4l. (Here stands for quotient; the rational numbers are those which are quotients of integers.)
We would now like to prove the well-known fact that the rational numbers form a field. Under the scheme that we have been using in this chapter, the proof involves an unexpected difficulty. Following a procedure which is the re-verse of the usual one, we have defined the positive integers in terms of the real numbers; and at the present stage we do not officially know that sums and products of integers are always integers. This can be proved, but we postpone the proof until the end of this chapter; in the meantime we regard the closure of the integers as a postulate.
CL. The Closure Postulate. The integers are closed under addition and multiplication.
The following theorem is now easy.
■
THEOREM 1. The rational numbers form an ordered field.PROOF. We shall verify the field postulates one at a time.
A-1. Closure Under Addition.
p +
r = p•s +—r _p•s + q•r = 1 ( p•s+q • r q s q-s s q•s q•s q•s p s+q• r)=q • s which is rational.
A-2. Since addition is associative for real numbers in general, it follows that addition is associative for rational numbers in particular.
(This is an instance of a general principle. If a postulate says that a certain equation is an algebraic identity, then this postulate automatically holds in any subsystem of the given system.)
A-3. Zero is rational, because 0 = 0/1.
A-4. Given a rational number p/q, we have —(p/q) = (—p)/q, which is rational.
A-5. Since addition is commutative for all real numbers, it is commutative for all rational numbers.
M-1. p/q
• r/s = pr/qs, which is rational.M-2. See the verifications of A-2 and A-5.
M-3. 1 is rational, because 1 = 1/1.
M-4. If p/q 0 0, then
p
0. Therefore (p/q)-1 = q/p, which is rational.M-5. See the verifications of A-2, A-5, and M-2.
AM-1. The distributive law holds for rational numbers, because it holds for all real numbers. ❑
30 The Algebra of the Real Numbers
Thus 0 forms a field. And the order relation <, given for all real numbers, applies in particular to the rational numbers, and the order postulates auto-matically hold.
Finally we remark, by way of preparation for some of the problems below, that if a number is rational, =