7.5: Exercises

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Mar 3, 2021
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- 7.4: Cardinality
- Back Matter

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1. List out the elements of the set “The letters of the word Mississipi”

2. List out the elements of the set “Months of the year”

3. Write a verbal description of the set $\{3,6,9\}$

4. Write a verbal description of the set $\{a, i, e, 0, u\}$

5. Is $\{1,3,5\}$ a subset of the set of odd integers?

6. Is $\{A, B, C\}$ a subset of the set of letters of the alphabet?

For problems 7-12, consider the sets below, and indicate if each statement is true or false.

$A=\{1,2,3,4,5\} \quad B=\{1,3,5\} \quad C=\{4,6\} \quad U=\{\text { numbers from } 0 \text { to } 10\}$

7. $3 \in B$

8. $5 \in C$

9. $B \subset A$

10. $C \subset A$

11. $C \subset B$

12. $C \subset U$

Using the sets from above, and treating $U$ as the Universal set, find each of the following:

13. $A \cup B$

14. $A \cup C$

15. $A \cap C$

16. $B \cap C$

17. $A^{c}$

18. $B^{c}$

Let $D=\{b, a, c, k\}, \quad E=\{t, a, s, k\}, \quad F=\{b, a, t, h\}$ . Using these sets, find the following:

19. $D^{c} \cap E$

20. $F^{c} \cap D$

21. $(D \cap E) \cup F$

22. $D \cap(E \cup P)$

23. $(F \cap E)^{c} \cap D$

24. $(D \cup E)^{c} \cap F$

Create a Venn diagram to illustrate each of the following:

25. $(F \cap E) \cup D$

26. $(D \cup E)^{c} \cap F$

27. $\left(F^{c} \cap E^{\prime}\right) \cap D$

28. $(D \cup E) \cup F$

Write an expression for the shaded region.

29.

$A Venn diagram with 3 circles overlapping, labeled A, B, and C. The region where B overlaps either or both of the other sets is highlighted.$

30.

$A Venn diagram with 3 circles overlapping, labeled A, B, and C. The region in B alone is highlighted, where it isn't overlapping either other set.$

31.

$A Venn diagram with 3 circles overlapping, labeled A, B, and C. The highlighted region includes all of C, combined with the portion of A that doesn't overlap B.$

32.

$A Venn diagram with 3 circles overlapping, labeled A, B, and C. The highlighted region includes the part of A that doesn't overlap with C, combined with the part of B that doesn't overlap C, combined with the overlap of all three circles.$

Let $A=\{1,2,3,4,5\} \quad B=\{1,3,5\} \quad C=\{4,6\}$ . Find the cardinality of the given set.

33. $\mathrm{n}(A)$

34. $\mathrm{n}(B)$

35. $\mathrm{n}(A \cup C)$

36. $\mathrm{n}(A \cap C)$

The Venn diagram here shows the cardinality of each set. Use this in 37-40 to find the cardinality of given set.

$A Venn diagram of three overlapping circles labeled A, B, and C. The part only in A is 7. The overlap of A and B only is 3. The part in B only is 5. The overlap of A and C only is 4. The overlap of all three is 1. The overlap of B and C only is 2. The part in C only is 8. The part outside all three is 6.$

37. $\mathrm{n}(A \cap C)$

38. $\mathrm{n}(B \cup C)$

39. $\mathrm{n}\left(A \cap B \cap C^{2}\right)$

40. $\mathrm{n}\left(A \cap B^{c} \cap C\right)$

41. If $n(G)=20, n(H)=30, n(G \cap H)=5,$ find $n(G \cup H)$

42. If $n(G)=5, n(H)=8, n(G \cap H)=4,$ find $n(G \cup H)$

43. A survey was given asking whether they watch movies at home from Netflix, Redbox, or a video store. Use the results to determine how many people use Redbox.

$\begin{array}{ll} \text{52 only use Netflix} & \text{62 only use Redbox} \\ \text{24 only use a video store} & \text{16 use only a video store and Redbox} \\ \text{48 use only Netflix and Redbox} & \text{30 use only a video store and Netflix} \\ \text{10 use all three} & \text{25 use none of these} \end{array}$

44. A survey asked buyers whether color, size, or brand influenced their choice of cell phone. The results are below. How many people were influenced by brand?

$\begin{array}{ll} \text{5 only said color} & \text{8 only said size} \\ \text{16 only said brand} & \text{20 said only color and size} \\ \text{42 said only color and brand} & \text{53 said only size and brand} \\ \text{102 said all three} & \text{20 said none of these} \end{array}$

45. Use the given information to complete a Venn diagram, then determine: a) how many students have seen exactly one of these movies, and b) how many had seen only Star Wars.

$\begin{array}{ll} \text{18 had seen The Matrix (M)} & \text{24 had seen Star Wars (SW)} \\ \text{20 had seen Lord of the Rings (LotR)} & \text{10 had seen M and SW} \\ \text{14 had seen LotR and SW } & \text{12 had seen M and LotR} \\ \text{6 had seen all three} & \text{} \end{array}$

46. A survey asked people what alternative transportation modes they use. Using the data to complete a Venn diagram, then determine: a) what percent of people only ride the bus, and b) how many people don’t use any alternate transportation.

$\begin{array}{ll} \text{30% use the bus} & \text{20% ride a bicycle} \\ \text{25% walk} & \text{5% use the bus and ride a bicycle} \\ \text{10% ride a bicycle and walk} & \text{12% use the bus and walk} \\ \text{2% use all three} & \text{} \end{array}$

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