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Unformatted text preview: 28 Chapter 1 Sets (d) D ={{1.2}.{1,2.3,4}}
(6) E = {15}
(f) F = {19.3.41} 1.4. Write each of the following sets by listing its elements within braces. 1.5. 1.6. 1.7. (a) A={nGZ: —4<n54} (b) B={neZ: n2<5} (c)C={rtEN: n3< 100} (d) D={xeR: xznx=0} (e) E={xeR: x2+1=0} Write each of the following sets in the form {x e Z : p(x)}, where p(x) is a property concerning x.
(a) A = {—1,—2,—3,...} (b) B = {—3,—2,...,3} (c) C = {—2, —l, 1,2} The set E = {2x : x e Z} can be described by listing its elements, namely E = {. . . , —4, ml, 0, 2, 4, . . .}.
List the elements of the following sets in a similar manner. (a) A={2x+1: er} (b) B={4n: n 62} (c) C={3q+l: qu} The set E = {. . . , —4, m2, 0, 2,4, . . .} of even integers can be described by means of a deﬁning condition
by E = {y = 2x : x e Z} = {2x : x e Z}. Describe the following sets in a similar manner. (a) A={...,—4,—l,2,5,8,...}
(b) B={...,—10,~5,0,5,10,...}
(c) C={l,8,27,64,125,...} Section 1.2: Subsets 1.8. 1.9. 1.10. 1.11. 1.12. 1.13. Give examples of three sets A, B, and C such that
(a) A E B C C. (b) A 63,3 eC,andA ¢C. (c) A eBandACC. Let (a, b) be an open interval of real numbers and let c e (a, 1:). Describe an open interval 1 centered at c
such that I Q (a, [9). Which of the following sets are equal?
A={neZ:n<2} D={neZ:n251}
B={nEZ:n3=n} E={—1,0,1}
C={nEZ: nzsn} For a universal set U = {1, 2, .. . , 8} and two sets A = {1. 3, 4, 7} and B = {4, S, 8}, draw a Venn diagram
that represents these sets. Find 73(A) and I'P(A) for (a) A = {1, 2}.
(b) A = [0, l, {a}}.
Find 'P(A) for A = {0, {0}}. 1.14. 1.15.
1.16. Exercises for Chapter 1 29 Find 79(’P({ l D) and its cardinality. Find PM) and P(A) for A = {0, £81, {{5}}.
Give an example of a set S such that (a) S S 73(N) (b) S e 73(N) (c) S E 'P(N) and Sl = 5. (d) S e ’P(N) and S = 5. Section 1.3: Set Operations 1.17. 1.18. 1.19.
1.20. 1.21. 1.22. 1.23. 1.24. Let U = {1, 3, .. . , 15} be the universal set, A = {1, 5, 9, 13}, and B = {3, 9, 15}. Determine the following:
(a)AUB, (mama, (c)A—B, (d)B—A, ((2)1 (arms, Give examples of three sets A, B, and C such that (a) AeB,AEC,andB§7__’C.
(b) BeA,BCC,andAﬂC9Eﬁ.
(c) AEB,B§C,andA,Q_C. Give examples of three sets A, B, and C such that B # C but B — A = C — A. Give examples of two sets A and B such that A — Bl = A 0 Bl = B — AI = 3. Draw the accompanying
Venn diagram. Let U be a universal set and let A and B be two subsets of U. Draw a Venn diagram for each of the
following sets. (a) m (b) in? (c) m (d) Eu? What can you say about parts (a) and (b)? parts (c) and ((1)? Give an example of a universal set U , two sets A and B, and an accompanying Venn diagram such that
AﬂBl=[A—B=B—A=[AUB=2. Let A, B, and C be nonempty subsets of a universal set U. Draw a Venn diagram for each of the following
set operations. (a) (C — B) U A (b) C n (A — B) Let A = {13. {121}. {[0}}}. (3) Determine which of the following are elements of A: [21, {ﬂ}, {1?}, {I}}. (b) Determine A . (0) Determine which of the following are subsets of A: G, {0}, {121, {6}}.
For (d)—(i), determine the indicated sets. (d) a n A (e) {a} n A (0 {EL {{5}} n A
(g) a u A (h) {a} u A (i) {a {an o A. 30 Chapter 1 Sets Section 1.4: Indexed Collections of Sets 1.25. 1.26. 1.27.
1.28. 1.29. 1.30. 1.31. Give examples of a universal set U and sets A, B, and C such that each of the following sets contains
exactlyoneelement: Ar‘lB ﬂC.(AﬂB)—C,(A DC)“ B,(B ﬂC)ﬁ A, A — (B UC),B ~(A UC),
C — (A U B), A U B U C. Draw the accompanying Venn diagram. For a real number 1‘, deﬁne Ar = {r2}, B, as the closed interval [r — 1, r + 1], and C, as the interval (r, 00).
For S = {1, 2, 4}. determine (a) Uaes Ar! and males A0.
(b) UaeS Bo, and mates 3,,
(C) UaES Co: and ﬂees Coe
Let A = {1, 2, 5}, B = {0, 2, 4}, C = {2, 3,4}, and S = {A, B, C}. Determine UXeSX and ﬂXeS X. For a real number 1', deﬁne 5,. to be the interval [1' — 1, r + 2]. Let A = [1, 3, 4}. Determine Um“ So, and ﬂaEA SW Let A = {a, b, .. . , 2} be the set consisting of the letters of the alphabet. For a: e A, let A“ consist of a and
the two letters that follow it, where A}. = {y, z, a} and A2 = {2, a, 1)}. Find a set S 5; A of smallest
cardinality such that UaES A,y = A. Explain why your set S has the required properties. For each of the following collections of sets, deﬁne a set A“ for each n e N such that the indexed collection
{A,,},,EN is precisely the given collection of sets. Then ﬁnd both the union and intersection of the indexed collection of sets.
(a) {[1, 2 +1),[1,2 +1/2),[1,2 + 1/3), ...}
(b) {(—1.2). (3/2.4). (5/3. 6), (—7/4. 8),   .} For each of the following, ﬁnd an indexed collection {Admin of distinct sets (that is, no two sets are equal)
satisfying the given conditions. (a) 0:11 An = {0} and U311 A” =[0,1].
(b) (13:, A, = {—1,0, 1] and [Jill A" = 2. Section 1.5: Partitions of Sets 1.32. 1.33. 1.34.
1.35. Which of the following are partitions of A = (a, b, c, d, e, f, g}? For each collection of subsets that is not a
partition of A, explain your answer. (a) S: = Maxie. g}. {19. f}. {d}} (b) S: = {1a, 5. Gd}, {6. f}} (C) 33 = {A} (d) 5'4 = {{a}, 0, {39. ad}, {3, f, g}} (e) 35 = {161.0%}. {5.8}: {6}. {5). fl} Which of the following sets are partitions of A = {1, 2, 3, 4, 5}?
(a) S: = {1113}!{2,5}} (’0) $2 = {11.2}: {3,495}. 12, 1}} (C) 33 = {11,2}.{2,31,13.4}.{4,5}} ((1) S4 = A Let A = {1, 2, 3, 4, 5, 6}. Give an example ofa partition S of A such that [S] = 3. Give an example of a set A with A = 4 and two disjoint partitions 51 and 82 of A with IS 1] =
[$21 = 3. . ._ _.__.._,__ ___ _ K _ _ 1.36. 1.37.
1.38. 1.39.
1.40. Additional Exercises For Chapter 1 31 Give an example of three sets A, S 1, and 82 such that S1 is a partition of A, $2 is a partition of $1, and
152] < [Sui < IAI Give an example of a partition of Q into three subsets.
Give an example of a partition of N into three subsets.
Give an example of a partition of Z into four subsets. Let A = {1, 2, . . . , 12}. Give an example of a partition S of A satisfying the following requirements:
(i) {SI 2 5, (ii) T is a subset ofS such that {Tl = 4 and J UXET XI = 10, and (iii) there is no element B e S
such that [B] = 3. Section 1.6: Cartesian Products of Sets 1.41.
1.42.
1.43.
1.44.
1.45.
1.46.
1.47. Let A = {x, y, z} and B = {.r, y}. Determine A x B. Let A = {1, {l}, {[1}}}. Determine A x A. For A 2 {51. (7]. Determine A x ’P(A). For A = {Q}, {m}. Determine A x ’P(A). For A = {1, 2} and B = {fl}, determine A x B and ’P(A) X 79(8). Describe the graph of the circle whose equation is x2 + y2 = 4 as a subset of R x R. List the elements of the set S = {(x, y) e Z X Z : {xl + {y} = 3}. Plot the corresponding points in the
Euclidean x y plane. ADDITIONAL EXERCISES FOR CHAPTER 1 REE—x 1.48. 1.49. 1.50. Let S = {—10, ~9, . . . , 9, 10}. Describe each ofthe following sets as {x e S : p(x)}, where p(x) is some
condition on x. (a) A ={~10,~9,...,—1,l,...,9, 10} (b) B 2 («10,—9, —I,0} (c) C = {—S,—4,...,7} (d) D ={—10,—9,...,4,6,7,..., 10} Describe each of the following sets by listing its elements within braces.
(a) {2:62; x3—4.r=0} (b) {x E R: Ix] = —1} (c) {meNz 2<m 55} (d) MEN: 05:153} (c) {keQ: k2—4=0} (f) {kelz 9k2—3z0} (g) {keZz 1518510} Determine the cardinality of each of the following sets.
(a) A = [1, 2, 3, {1. 2, 3}, 4, {4}} (b) B={xeR: lxlz—l} (c) C={meN: 2<m55} (d) D={neN: n<0} (e) E={kEN: 151135100} (f) F={keZ: 15185100} 32 Chapter 1 Sets 1.51. ForA = {—~1,0, I} and B = {x, y},determineA X B. 1.52. Let U = {1, 2, 3} be the universal set, and let A = {1,2}, B = {2, 3}, and C = {1, 3}. Determine the
following.
(a) (AUB)—(B QC)
(1)) A
(c) B UC
(d) A x B 1.53. Let A = {1, 2, .. ., 10}. Give an example of two sets S and B such that S 9 PM), IS] = 4, B e S, and
B = 2. 1.54. For A = {1} and C = {1,2}, give an example ofa set B such that 'P(A) C B C 79(C). 1.55. Give examples of two sets A and B such that
(a) A I“I’P(A) e B
(b) ’P(A) g A U B. 1.56. Which of the following sets are equal?
A={neZ:—4gn54} D={er:x=4x}
B={xeN:2x+2=0} E={ C={er: 3x—2=0} 1.57. Let A and B be sets in some unknown universal set U. Suppose that A = {3, 8, 9}, A — B = {1, 2},
B — A = {8}, and A (1 B = {5, 7}. Determine U, A, and B. 1.58. Let I denote the interval [0, 00). For each r e I, deﬁne 1 A,.={(x,y)eR><R: x2+y2=r2]
B,._—.{(x,y)eRxR: x2+y251‘2 ,
C,.={(x,y)eRxR: x2+y2>r2}. (a) Determine Ural A) and ﬁre, A,..
(1)) Determine Um, B, and ﬁre, 8,.
(e) Determine Um, C, and ﬁre, Cr.
1.59. Give an example of four sets Al, A2, A3, A; such that A, ['1 A Jl : {i — jl for everyr two integers i and j
with15£<js4.
1.60. (a) Give an example of two problems suggested by Exercise 1.59 (above).
(b) Solve one of the problems in (a).
1.61. Let A = {1, 2, 3}, B = {1, 2, 3, 4}, and C = {1, 2, 3, 4, 5}. For the sets S and T described below, explain
whether S < iTl, ISI > 1T,01' [31 = ITI
(a) Let B be the universal set and let S be the set of all subsets X of B for which X l 5A l'fl. Let T be the
set of 2~e1ement subsets of C. (b) Let S be the set of all partitions of the set A and let T be the set of 4element subsets of C.
(e) Let S = {(5,152) : b e B, a e A, a + bis odd} and let T be the set of all nonempty proper subsets of A. 1.62. Give an example of a set A = {1,2, . . . , k} for a smallest k e N having subsets A1, A2, A3 SuCh that
A, ——A,] = 1A, — Ail = {1' —j forevery two integersi andj with 1 5 t' < j 5 3. ...
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 Summer '10
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