Lista de exercícios 1 - Complementar - Gabarito parcial

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Universidade Federal de Santa Catarina
Centro de Cieˆncias F´ısicas e Matema´ticas
Departamento de Matema´tica
MTM3100 - Pre´-ca´lculo
Gabarito parcial da 1a lista complementar de exerc´ıcios
1. Ha´ mais de uma forma de representar um conjunto por propriedade.
(a) B = {x ∈ N | 4 ≤ x ≤ 9};(b)
(c) D = {x ∈ N | x ≥ 6};(d)
(e)
2. V;(a) (b) V;(c) (d)
F;(e) (f) V.(g)
3. A = {2, 3, 5, 7, 11};(a) (b) {2, 5};(c)
(d) {3, 7, 11};(e) (f)
4. A = {1,−1, 2,−2, 7,−7, 14,−14};(a) B = {. . . ,−9,−6,−3, 0, 3, 6, . . .};(b)
C = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29};(c) D = {2,−2, 3,−3, 5,−5, 7,−7};(d)
E = {2};(e) F = ∅;(f)
G = {a, r}.(g)
5. A = ∅;(a) B = {0, 2, 4, 6, . . .};(b)
C = {1, 3, 5, 7, . . .};(c) D = {0, 3, 6, 9, . . .};(d)
E = {−1, 4, 9, 14, 19, . . .};(e) F = {4, 5};(f)
G = {5, 6, 7};(g) H = {5};(h)
I = ∅;(i)
6. V;(a) (b) V;(c) (d)
F;(e) (f) F;(g) (h)
7. V;(a) (b) V;(c) (d)
V;(e) (f) F;(g) (h)
F;(i) (j) V;(k) (l)
F.(m)
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8. V;(a) F;(b) F;(c) F;(d)
V;(e) V;(f) V;(g) V.(h)
9. V;(a) V;(b) F;(c) V;(d)
F;(e) V;(f) V;(g) F;(h)
F.(i)
10. Ha´ 10 subconjuntos com treˆs elementos: {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5},
{2, 3, 5}, {2, 4, 5}, {3, 4, 5} e {2, 3, 4}.
11. (a) (b) (c)
(d) (e) (f)
12. (a) (b) (c) (d)
13. (1), (2), (3) e (5);(a) (1), (2), (4) e (7);(b)
(1) e (2);(c) (1) e (3);(d)
(1) e (4);(e) (1);(f)
(1), (2), (3), (4), (5) e (7);(g) (1), (2), (3), (4), (5) e (6);(h)
(1), (2), (3), (4), (6) e (7);(i) (1), (2), (3), (4), (5), (6) e (7);(j)
(1), (2), (3), (4), (5), (6), (7) e (8);(k) (4), (6), (7) e (8);(l)
(3), (5), (6) e (8);(m) (3), (4), (5), (6), (7) e (8);(n)
(2), (3), (5), (6), (7) e (8);(o) (7) e (8);(p)
(6) e (8);(q) (2), (3), (4), (5), (6), (7) e (8);(r)
(8).(s)
14. (2) e (8);(a) (3) e (4);(b)
(1) e (6);(c) (1), (3), (4) e (6);(d)
(5) e (7).(e)
15. Regia˜o (1): A ∩B ∩ C;
Regia˜o (2): A ∩B ∩ C;
Regia˜o (3): A ∩B ∩ C;
Regia˜o (4): A ∩B ∩ C;
Regia˜o (5): A ∩B ∩ C;
Regia˜o (6): A ∩B ∩ C;
Regia˜o (7): A ∩B ∩ C;
Regia˜o (8): A ∩B ∩ C.
16. Com 4 conjuntos havera´ 24 = 16 regio˜es. Para n conjuntos, havera´ 2n regio˜es.
17. 0.
2
18. A ∩D = {0, 2, 3, 5};(a) A ∩B = {0, 2, 4, 6, 8} = B;(b) A ∩ C = {1, 3, 7} = C;(c)
B ∩ C = ∅;(d) C ∩ ∅ = ∅;(e) B ∩D = {0, 2};(f)
A ∩ C ∩D = {3};(g) A ∩B ∩ C ∩D = ∅.(h)
19. A ∪D = {−2,−1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9};(a) (b)
A ∪ C = A;(c) (d)
B ∪ ∅ = B;(e) (f)
20. A−B = {1, 3, 5, 7, 9};(a) (b)
A− C = {0, 2, 4, 6, 8, 9};(c) (d)
A−D = {1, 4, 6, 7, 8, 9};(e) (f)
A− ∅ = A;(g) (h)
B − C = B;(i) (j)
21. {BA = A−B = {1, 3, 5, 7, 9};(a) (b)
{DA = A−D = {1, 4, 6, 7, 8, 9};(c) (d)
{CC = C − C = ∅;(e) (f)
{BD = D −B = {−2,−1, 3, 5};(g) (h)
{AC = C − A = ∅;(i)
22. A− (B ∪ C) = {5, 9};(a)
{(C∩A)A = {0, 2, 4, 5, 6, 8, 9};(b)
A− (B ∩ C) = A− ∅ = A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.(c)
23. B M D = {−2,−1, 3, 4, 5, 6, 8};(a) (B ∪D)− (B ∩D) = {−2,−1, 3, 4, 5, 6, 8}.(b)
B M D = (B ∪D)− (B ∩D) para quaisquer conjuntos.
24. B ∩ (C ∪D) = {0, 2};(a) (B ∩ C) ∪ (B ∩D) = {0, 2}.(b)
B ∩ (C ∪D) = (B ∩ C) ∪ (B ∩D) para quaisquer conjuntos.
25. A ∪ (B ∩D) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9};(a) (A ∪B) ∩ (A ∪D) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.(b)
A ∪ (B ∩D) = (A ∪B) ∩ (A ∪D) para quaisquer conjuntos.
26. A M B corresponde a` regia˜o
U
A B
3
27. A fo´rmula correta e´
n(A ∪B ∪ C) = n(A) + n(B) + n(C)− n(A ∩B)− n(A ∩ C)− n(B ∩ C) + n(A ∩B ∩ C).
28. 42;(a) (b)
29. 15;(a) (b) (c) (d)
30. {4, 5}; {0, 4, 5}; {4, 5, 6}; {0, 4, 5, 6}.
31. 239 subconjuntos.
32.
33. X = {1, 6, 9, 11};(a)
Y = {0, 2, 4, 7, 8};(b)
X ∩ Y = {3, 5, 10, 12, 13}.(c)
34. A ∩B = {2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13};(a) A ∪B = {2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13};(b)
A ∪B = {2, 3, 4, 10, 11, 12, 13};(c) A ∩B = {2, 3, 4, 10, 11, 12, 13}.(d)
A ∩B = A ∪B e A ∪B = A ∩B para quaisquer conjuntos A e B.
35.
B C
A
D
U
36. (a) (b) (c)
37. n(A ∩B ∩ C) = 2;(a) (b)
(c) n(A− (B ∩ C)) = 15.(d)
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