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Unformatted text preview: '\ i it; 28 Chopra0 Preliminaries . .l 7. Under what conditions does A \ (A\B) = B?
8. Show that (A\B) U (B\A) = (A U B)\(A n B).
9. Look up Russell’s paradox and write a brief summary discussing how it relates to Section 0.1.
10. Describe each of the following sets as the empty set, as R, or in interval notation, as appro
priate:  (a) m (__I_‘i)
n=l II n (b) ”g (en. n) (C) (3 (1.1 +1)
n=l n R
m 1 1 (d) "Lil (ﬁg, 2 + h) 11. Prove (ii) of Theorem 0.4.
12. Use De Morgan’s Laws to give a different and simpler description of the following sets: (a) Rt 0 (~1 1) v
":1 n H (b) C} (R\[i,2 +3) 0.2 RELATIONS AND FUNCTIONS 13. Deﬁne f :J«>be fol) = 2n « l for each n E J. What is imf? Is f1—1?Is fonto? If}c
has an inverse, find the domain of the inverse and give a formula for f"1(n). ' I 14. Often, if the domain is not speciﬁed, it is assumed to be the set of all real numbers for which
the formula for ﬁx) deﬁnes a real number. What is the domain of the function deﬁned by . the function f(x) = x
x + u 2? What is im f? Is 3” injective‘? If so, ﬁnd the inverse. For Exercises 15—17, MA = {1, 2, 3, 4, 5},B = {2, 3, 4, 5, 6, 7}, andC = {a, b, c, d, e}. 15. Give an example of f : A _., B that is not 1..1_
16. Give an example of f :A —> B that has an inverse, and show the inVcrse.
17. Give an example off : A ——> 3,313 ——> C such that g Of is 1—1 but g is not 11.'
*18. If f:A —> B is 1—1 and im 3“ = B, prove thatU"'1 0f)(a) = a for all a E A and
(ft1 1””) (b) = b for each 6 E B. 0.3 MATHEMATICAL INDUCTION AND RECURSION , 19. PIDVethatforallnEJ, 1 +247 +n=n(n; 1). 20. ProvethatforallnEJ,1E— 3 + 5 + + (2n w1)=rtz. 21. Prove that n3 + Sr: is divisible by 6 for each n E J. 22. Prove“that n2 < 2" for n E J, n a 5. (See Example 0.11.) 23. Prove the second principle of mathematical induction (Theorem 0.9). 24. Deﬁne f:J’—>be f0) = l, f(2) = 2, f(3) = 3, and ﬁn) = ﬁn — l) + ﬁn —' 2) + ‘ for — 3) for n 2 4. Prove that ﬂu) S 2” for all n E J. , ' 25. Deﬁnef : J—>be f(1) = 2 and, forn 2 2, ﬂit) 2 V3 + for — 1). Prove that ﬁn) < 2.4 for all n E J. You may want to use your calculator for this exercise. Project 0.] 29 26. Deﬁnef:J)be f(1) = 2, f9) = ~8. and, for n E 3, ﬁn) = 8f(n  1)—15f(n  2)
+ 6  2". Prove that, for all n E J, ﬂn) : —5  3” + 5"" ~l— 2””. 27. Prove Theorem 0.10. *28. Prove the following modiﬁed version of the second principle of mathematical induction: Let [’01) be a statement for each n E Z. If
(a) P010), Ptn0 + i), . . . , P(m) is true, and
(b) for k _>_ m, if PO) is true for n0 5 i S k, then P(k + 1) is true,
then P01) is true for all n 2 no, it E Z.  29. Deﬁne ﬁn) as follows for n E Z, n 2 0, f(0) = 7, ﬂ!) = 4, and, for n 2 2,
3°01) = 6f(n — 2) — ﬁn — I). Prove that 3°02) = 5 . 2" + 2(—3)” for all n E Z, n 2 t), 0.4 EQUIVALENT AND COUNTABLE SETS 30. Prove Corollary 0.15. 31. Find a 1—1 function J” from J onto 5 where S is the set of all odd integers. 32. Let P,. be the set of all polynomials of degree n with integer coefﬁcients. Prove that P" is
countable. (Hint: A proof by induction is one method of approach.) 33. Use Exercise 32 to show that the set of all polynomials with integer coefﬁcients is a countable
set. 34. Prove the following generalization of Theorem 0.17: If S is a countable set and {fixings is
an indexed family of countable sets, then UxeSA. is a countable set. 35. For each p E P,,, deﬁne B(p) = [x: p(x) = 0}. Prove that UpEP” B( p) is countable. 36. An algebraic number is any number that is a root of a polynomial equation p(.r) = 0 where
the coefﬁcients of p are integers. Show that the set of algebraic numbers is a countable set. 37. For a set A, let P04) be the set of all subsets of A. Prove that A is not equivalent to PM).
(Hint: Supposef : A —> PM) and deﬁne C = [x : x E A and x 6‘3 ffx) }. Show C 63 im 3‘.) 38. Let a, b, c, and d be any real numbers such that a < b and c < d. Prove that [a, b] is
equivalent to [c', d]. (Hint: Show that [a, b] is equivalent to [0, 1] ﬁrst.) 0.5 REAL NUMBERS + r
”‘39. If): < y, prove thatx < x y < y.
+
*40. If x a 0 and y 2 0, prove that ny S “lg—X (Hint: Use the fact that (V— — V502 22 0.)
“‘41 If{_)<a<b,provethat0<a2<b2and0<\/E<VE
+
42. If x, y, a. and b are greater than zero and? < 3' prove that; < : + 2 E.
43. Let A = [r : r is a rational number and r2 < 2}. Prove the A has no largest member. (Hint:
. t __ 2
If r2 < 2, and r > 0, choose a rational number 5 such that 0 <, 6 < l and 6 < 2 +2 .
r PROJECT 0.1 . I
_ The purpose of this project is to show that the open interval (0, '1) is equivalent to the
closed interval {0. 1]. In the process we will discover that both intervals are equivalent
to [0, 1) and (0, I}. It is then easy to generalize to any interval [0, b] with a < b. 54 Chapter] Sequences and the statement holds for n = r + 1. By induction, the statement holds for all n, and
{a,,];;1 converges to 2. I I _ Example 1.10 Recall now a word of philosophy mentioned earlier—namely, that
determining the limit of a sequence may be half the battle in showing a sequence to be
convergent. Consider the sequence {Whisk Let us try to guess the limit in advance
by reasoning similar to that used in the preceding paraggg h. Suppose {Whig con~
verges; call the limit L. Now consider the subsequence { 2nlfg1; W= Wit/E, and we know {Vi}:=, converges to 1 (see Exercise 38). Thus, {Wlff'ﬁl converges to L and also to W; hence, by arguments given before, L = 1. We shall to prove
that the sequence converges to l or, equivalently, that the sequence [ n _— l}:=1
converges to 0. Let x,, = W  1; clea'riy x" 2 0 and — — 1
n=(1+x,,)”= l+nxn+§££2—Dxi+1u+x:2n(n—22xﬁ.
Thus, for all n .2 2, we have .0 5 xn 5 V2] n  1). It should be clear now how to
complete the proof that {x,,};',°=1 converges to 0. , I
EXERCISES 1.1 ‘SEQUENCES AND CONVERGENCE
1. Show that [0, 1] is a neighborhood of ﬁe—that is, there is e > 0 such that 2 2
(3—6.3+6)C[0,l]. *2. Let x and y be distinct tea] numbers. Prove there is a neighborhood P of x and a neighborhood
Qofy such thatP n Q = (2). *3. Suppose I is a real number'and s > 0. Prove that (x — e, x + E) is a neighborhood of
each of its members; in other words, if y E (x — s, x + s), then there is 8 > 0 such that (y—8,y+8)C(xe,x+€).
3n+7}‘° ﬂ n=l ’
5. Give an example of a sequence that is bounded but not convergent.
6. Use the deﬁnition of convergence to prove that each of the following sequences converges: (a) {5 + l}
n ‘ ’1 n=1 4. Find Upper and lower bounds for the sequence { Exercises 55 (C) {2_"}:=1_ e: 3n
(1 _.._
,( ) 2n + 1 ”=5
*7. Show that {an};°=l converges toA iff {.51,1  ALL. converges to 0.
8. Suppoae {cum} converges to A, and deﬁne a new sequence {bah}; by b" = a" 4‘20"“ for all n. Prove that {.b,I };;1 converges to A. Suppose {anﬁLh {b.h‘fsl, and {cu};‘;1 are sequences such that {an};',°=1 converges to A, {£5};le converges to A, and a“ s c,1 5 Inn for all n. Prove that {c,I 3;] converges to A. *10. Prove that, if {an} i=1 converges to A, then { IanI };°=, converges to IA I. Is the converse true?
Justify your conclusion. *11. Let {an} 22.1 be a sequence such that there exist numbers a: and N such that, for n a N, an = a. Prove that {an}:=; converges to a. 12. Give an alternate proof of Theorem 1.1 along the following lines. Choose 6 > 0. There is
N1 such that fern 2. N1 Ian —. AI < g, and there is N2 such that for n 2 N2 Ian  BI < 5.
Use the triangle inequality to show that this implies mat IA — BI < e. Argue that A = B. 13. Let x be any positive real number, and deﬁne a sequence {an}:=] by *9 =[x]+[2xl++lnxi If)
:1 H; where [x] is the largest integer lees than or equal to x. Prove that Ia,1 In=1 converges to x12. 1.2 CAUCHY SEQUENCES l4. Prove that every Cauchy sequence is bounded (Theorem 1.4). 15. Prove directly (do not use Theorem 1.8) that, if {chitin} and [17.52.] are Cauchy, so is
{“11 + bu}:z=l 16. Prove directly (do not use Theorem 1.9) that, if {anlfsl and {bniffgl are Cauchy, so is
{ anbn] 3:1. You will want to use Theorem 1.4. 2n+1 “'
n n=!
18. Give an example of a set with exactly two accumulation points.
19. Give an example of a set with a countably inﬁnite set of accumulation points.
20. Give an example of a set that contains each of its accumulation points.
21. Determine the accumulation points of the set {2" + i: n and k are positive integers}.
‘22. Let S be a nonempty set of real numbers that is bounded from above (below) and let
x = sup S (inf S). Prove that either x belongs to S or x is an accumulation point of S.
artl. + arr’2 1'7. Prove that the sequence is Cauchy. 23. Let an and a, be distinct real numbers. Deﬁne a,r = for each positive integer 2
rt 2 2. Show that {an}:'._.1 is a Cauchy sequence. You may want to use induction to show
that
a a — “1— n(a —a)
n n+1 I! 2 1 0 and then use the result from Example 0.9 of Chapter 0. ...
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 Fall '07
 RickRugangYe

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