15.1: Chapter 1

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Your Turn

1.1

1. One possible solution: /**/T = \{ {\text{wrench, screwdriver, hammer, plyers}}\}/**/.

1.2

1. This is not a well-defined set.

2. This is a well-defined set.

1.3

1. /**/\emptyset {\text{ or \{ \} }}/**/

1.4

1. /**/D = \{ 0,1,2, \ldots ,9\} /**/

1.5

1. /**/M = \{ 1,3,5, \ldots \} /**/

1.6

1. /**/C = \{c|c{\text{ }}{\text{is a car}}\}/**/

1.7

1. /**/I = \{i|i{\text{ }}{\text{is a musical instrument}}\}/**/

1.8

1. /**/n(P) = 0/**/

2. /**/n(A) = 26/**/

1.9

1. finite

2. infinite

1.10

1. Set /**/B/**/ is equal to set /**/A/**/, /**/B = A/**/

2. neither

3. Set /**/B/**/ is equivalent to set /**/C/**/, /**/B \sim C/**/

1.11

1. /**/\{ {\text{heads, tails}}\} ;/**//**/\{ {\text{heads}}\} ,{\text{ }}\{ {\text{tails}}\} ;/**/ and /**/\emptyset /**/

1.12

A set with one member could contain any one of the following:

/**/\{ {\text{Articuno}}\} {\text{, }}\{ {\text{Zapdos}}\} {\text{, }}\{ {\text{Moltres}}\} {\text{, or }}\{ {\text{Mewtwo}}\} /**/.

Any of the following combinations of three members would work:

/**/\{ {\text{Articuno, Zapdos, Mewtwo}}\} /**/,/**/\{ {\text{Articuno, Moltres, Mewtwo}}\} /**/, or /**/\{ {\text{Zapdos, Moltres, Mewtwo}}\} /**/.

3. The empty set is represented as /**/\{ {\text{ }}\} /**/ or /**/\emptyset /**/.

1.13

1. /**/E \subset \mathbb{N}/**/

1.14

1. 512

1.15

1. /**/\{ m|m = 5n{\text{ where }}n \in \mathbb{N}\}/**/

1.16

Serena also ordered a fish sandwich and chicken nuggets, because for the two sets to be equal they must contain the exact same items: {fish sandwich, chicken nuggets} = {fish sandwich, chicken nuggets}.

1.17

There are multiple possible solutions. Each set must contain two players, but both players cannot be the same, otherwise the two sets would be equal, not equivalent. For example, {Maria, Shantelle} and {Angie, Maria}.

1.18

1. The set of lions is a subset of the universal set of cats. In other words, the Venn diagram depicts the relationship that all lions are cats. This is expressed symbolically as /**/{L} \subset {U}/**/.

1.19

1. The set of eagles and the set of canaries are two disjoint subsets of the universal set of all birds. No eagle is a canary, and no canary is an eagle.

1.20

1. The universal set is the set of integers. Draw a rectangle and label it with /**/U = \text{Integers}/**/. Next, draw a circle in the rectangle and label with Natural numbers.

A Venn diagram shows a circle placed inside a rectangle. The circle represents natural numbers and is shaded in yellow. The rectangle represents U equals integers and is shaded in blue.

Venn diagram with universal set, /**/U=\text{Integers}/**/, and subset /**/\mathbb{N} = {\text{Natural numbers}}/**/.

A Venn diagram shows a circle placed inside a rectangle. The circle represents A and is shaded in yellow. The rectangle represents U and is shaded in blue.

Venn Diagram with universal set, /**/U/**/ and subset /**/A/**/.

1.21

A Venn diagram shows two circles are placed inside a rectangle. The circle on the left represents birds and is shaded in orange. The circle on the right represents airplanes and is shaded in yellow. The rectangle represents U equals things that can fly and is shaded in blue.

Venn Diagram with universal set, /**/U =/**/ Things that can fly with disjoint subsets Airplanes and Birds.

1.22

1. /**/A' = \left\{

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2. /**/A' = \{ c \in U|c{\text{ is a lion}}\}/**/ or /**/A' = \{ c \in U|c \notin A\}/**/

1.23

1. /**/A\mathop \cap \nolimits B = \{ a\}/**/

1.24

1. /**/A \cap B = \{\,\,\}/**/

1.25

1. /**/A \cap B = B = \{a,e,i,o,u\}/**/

1.26

1. /**/A \cup B = \{a,d,h,p,s,y\}/**/

1.27

1. /**/A \cup B = \{{\text{red, yellow, blue, orange, green, purple\}}}/**/

1.28

1. /**/A \cup B = \{a, b, c,\ldots,z\} = A/**/.

1.29

1.30

113

1.31

1. /**/A/**/ or /**/B = A \cup B = \{h, a, p, y, w, e, s, o, m\} ./**/

2. /**/A/**/ and /**/C = A \cap C = \{a, h\} ./**/

3. /**/B/**/ or /**/C = B \cup C = \{a, w, e, s, o, m, t, h\} ./**/

4. (/**/A/**/ and /**/C/**/) and /**/B = (A \cap C) \cap B = \{a, h\} \cap \{a, w, e, s, o, m\} = \{a\} ./**/

1.32

127

1.33

/**/A \cap B = \{ 3,5,7\}/**/.

/**/A \cup B = \{ 1,2,3,5,7,9\} /**/.

/**/A \cap B' = \{ 1,9\} /**/.

/**/n(A \cap B') = 2/**/.

1.34

1. 40

2. 0

3. 27

1.35

/**/n(B) = n(A{B^ + }) + n(A{B^ - }) + n({B^ + }) + n({B^ - }) = 14/**/

/**/n({B^\prime }) = n(U) - n(B) = 86/**/

/**/n(B \cup R{h^ + }) = 87/**/

1.36

A Venn diagram shows three intersecting circles inside a rectangle. The first circle representing soup is shaded in yellow and has a value of 4. The second circle representing the sandwich is shaded in red and has a value of 15. The third circle representing salad is shaded in blue and has a value of 8. The intersecting region of soup and sandwich has a value of 2. The intersecting region of soup and salad has a value of 3. The intersecting region of sandwich and salad has a value of 10. The intersecting region of all three circles has a value of 8. The rectangle represents U equals conference attendees equals 50 and has a value of 0.

Venn diagram – Attendees at a conference with sets: Soup, Sandwich, Salad – Complete Solution

1.37

1. /**/A \cap (B \cap C) = \{ 0,1,2,3,4,5,6\} \cap \{ 0,6,12\} = \{ 0,6\}/**/

2. /**/(A \cap B) \cup (B \cap C) = \{ 0,2,4,6\} \cup \{ 0,3,6\} = \{ 0,2,3,4,6\}/**/

3. /**/(A \cup {C^\prime }) \cap (B \cup {C^\prime }) = \{ 0,1,2,3,4,5,6,7,8,10,11\} \cap \{ 0,1,2,4,5,6,7,8,10,11,12\} = \{ 0,1,2,4,5,6,7,8,10,11\}/**/

1.38

The left side of the equation is:

Two Venn diagrams. The first diagram represents A intersection B. It shows two intersecting circles A and B placed inside a rectangle. The rectangle represents U. The intersecting region of the two circles is shaded in blue. The second diagram represents the complement of A intersection B. It shows two intersecting circles A and B placed inside a rectangle. The rectangle represents U. Except for the intersecting region, the other regions of the two circles are shaded in blue.

Venn diagram of intersection of two sets and its complement.

The right side of the equation is given by:

Three Venn diagrams. The first diagram represents A complement. A rectangle U with a circle A on its left. The region inside the rectangle, outside the circle, is shaded in blue. The second diagram represents a B complement. A rectangle U with a circle B on its right. The region inside the rectangle, outside the circle, is shaded in yellow. The third diagram represents A complement union B complement. Two intersecting circles A and B are placed inside a rectangle. The rectangle represents U. Except for the region of intersection, all other regions of the two circles are shaded in green.

Venn diagram of union of the complement of two sets.

Check Your Understanding

1. set

2. cardinality

3. not a well-defined set

4. 12

5. equivalent, but not equal

6. finite

Roster method: /**/\{ \text{A, B, C, } \ldots \text{, Z}\},/**/ and set builder notation: /**/\{ x|x{\text{ is a capital letter of the English alphabet}}\}/**/

8. subset

To be a subset of a set, every member of the subset must also be a member of the set. To be a proper subset, there must be at least one member of the set that is not also in the subset.

10. empty

11. true

12. /**/{2^{10}} = 1024/**/

13. equivalent

14. equal

15. relationship

16. universal

17. disjoint or non-overlapping

18. complement

19. disjoint

20.

intersection

21.

union

22.

/**/A \cup B/**/

23.

/**/A \cap B/**/

24. /**/A/**/

25. /**/B/**/

26. empty

27. /**/n(A \cup B) = n(A) + n(B) - n(A \cap B)/**/

28.

overlap

29.

central

30. intersection of all three sets, /**/A \cap B \cap C/**/

31.

parentheses, complement

32. equation, true

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