8.2: Sets and Counting
- Page ID
- 192506
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Sets, Set Operations, Venn Diagrams, and Cardinality
1) \(\{m, i, s, p\}\)
3) One possibility is: Multiples of 3 between 1 and 10
5) Yes
7) True
9) True
11) False
13) \( A \cup B = \{1,2,3,4,5\} \)
15) \( A \cap C = \{4\} \)
17) \( A^\text{C} = \{6,7,8,9,10\} \)
19) \( D^\text{C} \cap E = \{t,s\} \)
21) \( (D \cap E ) \cup F = \{a,b,h,k,t\} \)
23) \( ( F \cap E )^\text{C} \cap D = \{b,c,k\} \)
25) 
27) 
29) One possible answer: \( ( A \cap B ) \cup ( B \cap C ) \)
31) \( \left( A \cap B^\text{C} \right) \cup C \)
33) 5
35) 6
37) \( n( A \cap C ) = 5 \)
39) \( n( A \cap B \cap C ) = 3 \)
41) \( n( G \cup H ) = 45 \)
43) 136 use Redbox
45) 
a) 8 had seen exactly one
b) 6 had only seen SW
Tree Diagrams and the Multiplication Axiom
1) 6
3) 8
5) 12
7) 15,600,000
9) 6,400,000
11) BB, BG, GB, GG
13) 16
15) 27,000
17) 10,000
Permutations
1) 60
3) 210
5) 362,880
7) 25,200
9) 900
11) 48
13) 72
15) 2,400
17) 15,120
19) 5,040
21) 720
Combinations
1) 120
3) 10
5) 2,598,960
7) 66
9) 10
11) 20
13) 6
15) 924
17) 35
Combinations Involving Several Sets
1) 24
3) 25
5) 14,400
7) 4
9) 60
11) 80
13) 51
15) 7
17) 1,410
19) 171,600
21) 22,308
23) 24
25) 858,627,790,020
27) 11,647,387,752,000
29) 1890
31) 1680
Circular Permutations and Permutations with Similar Elements
1) 24
3) 120
5) 120
7) 64,864,800
9) 210
11) 6
13) 10
15) 210
17) 15,120
19) 5,040
21) 720
Chapter Review
1) 1,000
2) 20; 135; 15
3) 12
4) 144
5) 3,024
6) 11,639,628,000
7) 84
8) 60
9) 24
10) 126; 336; 210
11) 5,184
12) 1,048,576
13) 46,200
14) 60
15) 120
16) 20
17) 10
18) 1,296
19) 27,720
20) 720
21) 194,594,400
22) a) 5,148; b) 58,656; c) 123,552; d) 10,240 or 9,216
23) 17,576
24) 4,500
25) 5,040; 720
26) 3003; 371; 210; 191; 435
27) 10
28) 35
29) 72
30) 72,000
31) -48384\(x^5y^3\)
32) 2016\(a^5b^4\)