212 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6(b) A subset S of R is said to be closed under addition if and only if x + y E S
whenever x E S and y E S. Suppose the subset S of R is closed under addition
and has the further property that -x E S whenever x E S. Prove that if x E S and
y 4: S, then x + y $ S. (Note: Z and Q both satisfy the hypotheses of this theorem.)
(c) A subset S of R is said to be closed under multiplication if and only if xy E S
whenever x E S and y E S. Suppose the subset S of R is closed under multipli-
cation and has the further property that l/x E S whenever x E S and x # 0. Prove
that if x E S, x # 0, and y 4 S, then xy 4 S. (Note: Q satisfies the hypotheses of
this theorem.)
*(d) Use the result of Example 9, together with the result of (c), to show that, for any
real number x, either & + x or ,h - x is irrational.
(e) Assume it is known that ,h is irrational whenever the positive integer n is not
a perfect square. Use this fact to prove that & + & is irrational.
- (a) [Continuation of Exercise 10(c), Article 6.11 Use proof by contrapositive to
show that if a subset S of R is open, then its complement S' is closed.
(b) Use (a) and Exercise 8(e), Article 6.1, to show that 0 is a closed subset of R.
*(c) Prove that 0 is an open subset of R.
(d) Prove that (25 is an interval in R. (Hint: Using the definition from Example 2,
Article 5.2, consider what must be the case if 0 is not an interval.) - Suppose that S is a linearly dependent subset of a vector space V and that
S E T. Prove that T is linearly dependent [recall Exercise lqa), Article 5.21. - Prove that if x and y are real numbers with y I x + p for every p > 0, then y I x.
- Suppose {x,} and {y,} are infinite sequences of real numbers such that x, + x,
y, -+ y, and x, < y, for all n = 1,2,3,.... Prove that x I y. Is it possible to prove
x < y? (Recall Exercise 20, Article 6.1.) - Each of the following theorems was proved earlier in the text (either as an exer-
cise or example) by direct methods. For each of them, set up a proof by contra-
positive and compare the effectiveness of this approach with that of the corresponding
direct proof.
(a) If m, n, and p are integers such that m In and mlp, then ml(n + p).
*(b) If A, B, and C are sets with A E B and B G C, then A c C.
(c) IfA,B,andXaresetswithAnXcBnXandAnX'cBnX',thenA~B. - Suppose f and g are both functions defined on an open interval containing a,
where a is a real number. Prove that if lirn,,, f(x) = L # 0 and lirn,,, g(x) = 0,
then lirn,,, f(x)/g(x) does not exist (recall Exercise 18, Article 6.1).
Existence and Uniqueness (Optional)
In this article we consider how to prove two types of mathematical theorem:- Prove that at least one object exists satisfying a given mathematical
property (existence). - Prove that at most one object exists satisfying a given mathematical
property (uniqueness).