15.0: Chapter 1

Last updated
Save as PDF

Page ID: 129930

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

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\{

ParseError: invalid DekiScript (click for details)

Callstack:
    at (Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/15:_Answer_Key/15.00:_Chapter_1), /content/body/div[3]/div[1]/span, line 1, column 1

\right\}\)

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.

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 intersection of two sets and its complement.

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

Search

Text Color

Text Size

Margin Size

Font Type