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Sets Exercises
Acknowledgment: All course slides are either referenced to Rosen Book online presentations (with certain amendments)
or are personally developed by the instructor.
Exercise 2 pp 131 @[KB 8ed]
2. Use set builder notation to give a description of each of these sets.
a) {0, 3, 6, 9, 12}
b) {−3,−2,−1, 0, 1, 2, 3}
c) {m, n, o, p}
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Exercise 4 pp 131 @[KB 8ed]
4. For each of these intervals, list all its elements or explain why it is
empty.
a) [a, a]
b) [a, a)
c) (a, a]
d) (a, a)
e) (a, b), where a > b
f) [a, b], where a > b
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Exercise 10 pp 132 @[KB 8ed]
10. For each of the following sets, determine whether {2} is an element
of that set.
a) {x ∈ R | x is an integer greater than 1}
b) {x ∈ R | x is the square of an integer}
c) {2,{2}}
d) {{2},{{2}}}
e) {{2},{2,{2}}}
f) {{{2}}}
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Exercise 12 pp 132 @[KB 8ed]
12. Determine whether these statements are true or false.
a) ∅ ∈ {∅}
b) ∅ ∈ {∅, {∅}}
c) {∅} ∈ {∅}
d) {∅} ∈ {{∅}}
e) {∅} ⊂ {∅, {∅}}
f) {{∅}} ⊂ {∅, {∅}}
g) {{∅}} ⊂ {{∅}, {∅}}
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Exercise 14 pp 132 @[KB 8ed]
14. Use a Venn diagram to illustrate the subset of odd integers in the
set of all positive integers not exceeding 10.
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Exercise 16 pp 132 @[KB 8ed]
16. Use a Venn diagram to illustrate the relationship A ⊆ B and B ⊆ C.
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Exercise 20 pp 132 @[KB 8ed]
20. Find two sets A and B such that A ∈ B and A ⊆ B.
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Exercise 26 pp 132 @[KB 8ed]
26. Determine whether each of these sets is the power set of a set,
where a and b are distinct elements.
a) ∅
b) {∅, {a}}
c) {∅, {a}, {∅, a}}
d) {∅, {a}, {b}, {a, b}}
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Exercise 32 pp 132 @[KB 8ed]
32. Suppose that A x B = ∅, where A and B are sets. What can you
conclude?
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Exercise 40 pp 133 @[KB 8ed]
40. Show that A x B ≠ B x A, when A and B are nonempty, unless A = B.
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Exercise 42 pp 133 @[KB 8ed]
42. Explain why (A x B) x (C x D) and A x (B x C) x D are not the same.
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Exercise 44 pp 133 @[KB 8ed]
44. Prove or disprove that if A, B, and C are nonempty sets and
A x B = A x C, then B = C.
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Exercise 4 pp 144 @[KB 8ed]
4. Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find
a) A ∪ B.
b) A ∩ B.
c) A − B.
d) B − A.
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Exercise 8 pp 144 @[KB 8ed]
8. Prove the idempotent laws in Table 1 by showing that
a) A ∪ A = A.
b) A ∩ A = A.
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Exercise 10 pp 144 @[KB 8ed]
10. Show that
a) A − ∅ = A.
b) ∅ − A = ∅.
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Exercise 14 pp 144 @[KB 8ed]
14. Find the sets A and B if A − B = {1, 5, 7, 8}, B − A ={2, 10}, and
A ∩ B = {3, 6, 9}.
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Exercise 20 pp 144 @[KB 8ed]
20. Let A, B, and C be sets. Show that
a) (A ∪ B) ⊆ (A ∪ B ∪ C).
b) (A ∩ B ∩ C) ⊆ (A ∩ B).
c) (A − B) − C ⊆ A − C.
d) (A − C) ∩ (C − B) = ∅.
e) (B − A) ∪ (C − A) = (B ∪ C) − A.
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Exercise 26 pp 145 @[KB 8ed]
Let A, B, and C be sets. Show that (A − B) − C = (A − C) − (B − C).
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Exercise 32 pp 145 @[KB 8ed]
32. Can you conclude that A = B if A, B, and C are sets such that
a) A ∪ C = B ∪ C?
b) A ∩ C = B ∩ C?
c) A ∪ C = B ∪ C and A ∩ C = B ∩ C?
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Exercise 36 pp 145 @[KB 8ed]
36. Prove or disprove that for all sets A, B, and C, we have
a) A x (B ∪ C) = (A x B) ∪ (A x C).
b) A x (B ∩ C) = (A x B) ∩ (A x C).
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Exercise 52 pp 145 @[KB 8ed]
52. Show that if A, B, and C are sets, then
|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|.
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