6.2 INDIRECT PROOFS 211(ii) Prove, by the choose method, that if A, B, and C are sets, then
A u (B n C) E (A u B) n (A u C).
(c) Describe the general approach you would take to prove these theorems:
(i) If a subset S of R is compact, then S is closed and bounded.
(ii) If A and B are sets such that A u B is finite, then A is finite and B
is finite.
(iii) The subset 22 of R consisting of all even integers is closed under
addition multiplication.[Note: It is not necessary to know the meaning of technical terms, such as
"bounded," in order to do (c).]- (a) Verify the tautology
[(PI + 41b(P2 -+ q2)I -+ [(PI vp2) -, (q1 v q2)I
(b) Use the result in Example 1 to show that if x satisfies the equation
x2 + 2x - 35 = 0, then either x = 5 or x = -7.
(c) Prove that if x, a, and b are real numbers satisfying the equation xu = xb, then
either x = 0 or a = b. (Use the result of Example 1.) - (a) Give a proof by contrapositive that if a function f (mapping R to R) is
increasing, then f is one to one.
(b) Prove that if f(x) = Mx + B is one to one, then M # 0. - (a) Prove that if m is an integer such that m2 is even, then m is even.
(b) Prove that if m is an integer such that m2 is odd, then m is odd [recall
Exercise 3(b), Article 6.11. - It is a familiar property (transitivity) of equality of real numbers that if a = b
and b = c, then a = c. Use this property to show that iff, g, and h are real-valued
functions of a real variable with A = (x E R I f(x) # g(x)}, B = (x E R lg(x) # h(x)),
and C = (x E Rl f(x) # h(x)}, then C G A u B.
*8. [Continuation of Exercise 16(c), Article 6.11 Let f be a real-valued function
defined on an open interval (a, b). Suppose f has a relative maximum at a point
x, E (a, b); that is, suppose there exists 6 > 0 such that f(x,) > f(x) for every
x E(X~ - 6, X, + 8). Suppose finally that f '(x,) exists. Prove that f '(x,) = 0.
- (a) Verify, by using a truth table, that [(p A q) + r] t, [(p A -- r) + -- q] is a
tautology.
(b) Suppose it is known that the sum of two rational numbers is rational. Prove
that if x is rational and x + y is irrational, then y is irrational. Can we conclude
y is irrational if we know that x and x + y are irrational?
*(c) Given sets A, B, and C such that A is a subset of B and A is not a subset of C.
Prove that B is not a subset of C.
(d) If m, n, and p are integers such that m divides n and m does not divide p, prove
that n does not divide p.
(e) Prove that if S, and S2 are subsets of R such that S, is convex and S, n S, is
not convex, then S, is not convex (recall Exercise 9, Article 5.2). - (a) Prove or disprove: If real numbers x and y are irrational, then x + y is
irrational? xy is irrational?