Elementary Abstract Algebra- Examples and Applications, 2019a

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5.3 DO THE SUBSETS OF A SET FORM A GROUP? 193 Although Exercise 5.3.2 deals with a particular set of subsets, the results of the exercise are completely general and apply to the set of any subsets of any set (and not just {a, b, c}. Now we’ll consider ∩: Exercise 5.3.3. Given a set A, letG be the set of all subsets of A. (a) Does the set G with the operation ∩ have the closure property? Justify your answer. (b) Does the set G with the operation ∩ have an identity? If so, what is it? Which part of Proposition 5.2.3 enabled you to draw this conclusion? (c) Is the operation ∩ defined on the set G associative? Proposition 5.2.3 enabled you to draw this conclusion? ♦ Which part of (d) Is the operation ∩ defined on the set G commutative? Which part of Proposition 5.2.3 enabled you to draw this conclusion? (e) Does each element of G have a unique inverse under the operation ∩? If so, which part of Proposition 5.2.3 enabled you to draw this conclusion? If not, provide a counterexample. (f) Is the set G a group under the ∩ operation? Justify your answer. No doubt you’re bitterly disappointed that neither ∩ nor ∪ can be used to define a group. However, take heart! Mathematicians use these operations to define a different sort of algebraic structure called (appropriately enough) a Boolean algebra. We won’t deal further with Boolean algebras in this course: suffice it to say that mathematicians have defined a large variety of abstract algebraic structures for different purposes. Although ∩ and ∪ didn’t work, there is a consolation prize: Exercise 5.3.4. Besides ∪ and ∩, there is another set operation called symmetric difference, which is sometimes denoted by the symbol Δ and is defined as: AΔB =(A \ B) ∪ (B \ A). Given a set U, letG be the set of all subsets of U. Repeat parts (a)–(f) of Exercise 5.3.3, but this time for the set operation Δ instead of ∩. ♦ ♦
194 CHAPTER 5 SET THEORY 5.4 Hints for “Set Theory” exercises Exercise 5.1.3(a): A century is a collection of years, . . . . Exercise 5.3.1(d): Guess the pattern from the previous parts of this exercise.
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Abstract Algebra: Examples and Appl
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2 Material from the 2012 version of
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Contents Forward 1 To the student:
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6 CONTENTS 3 Modular Arithmetic 90
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8 CONTENTS 6.7.1 Concept and defini
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10 CONTENTS 10 Symmetries of Plane
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12 CONTENTS 14 Equivalence Relation
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14 CONTENTS 18 Homomorphisms of Gro
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16 CONTENTS 21.8 Non-formula induct
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2 CONTENTS But students who don’t
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4 CONTENTS • The book’s web sit
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Organization Plan of the Book A cha
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8 CONTENTS connect algebraic aspect
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Glossary of Symbols cis θ: cosθ +
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1 Preliminaries 1.1 In the Beginnin
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14 CHAPTER 1 PRELIMINARIES multipli
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16 CHAPTER 1 PRELIMINARIES (a) (((x
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18 CHAPTER 1 PRELIMINARIES (b) a
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20 CHAPTER 1 PRELIMINARIES think ba
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22 CHAPTER 1 PRELIMINARIES Reason P
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24 CHAPTER 1 PRELIMINARIES Exercise
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2 Complex Numbers horatio: O day an
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28 CHAPTER 2 COMPLEX NUMBERS • In
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30 CHAPTER 2 COMPLEX NUMBERS 2.1.2
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32 CHAPTER 2 COMPLEX NUMBERS ♦ Pr
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34 CHAPTER 2 COMPLEX NUMBERS follow
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36 CHAPTER 2 COMPLEX NUMBERS What c
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38 CHAPTER 2 COMPLEX NUMBERS Exerci
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40 CHAPTER 2 COMPLEX NUMBERS (a) (3
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42 CHAPTER 2 COMPLEX NUMBERS Additi
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44 CHAPTER 2 COMPLEX NUMBERS so tha
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46 CHAPTER 2 COMPLEX NUMBERS (a) Sh
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50 CHAPTER 2 COMPLEX NUMBERS We kno
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52 CHAPTER 2 COMPLEX NUMBERS (a) Fi
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54 CHAPTER 2 COMPLEX NUMBERS ♦ We
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56 CHAPTER 2 COMPLEX NUMBERS (a) 2
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58 CHAPTER 2 COMPLEX NUMBERS If (r
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64 CHAPTER 2 COMPLEX NUMBERS ♦ Ex
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66 CHAPTER 2 COMPLEX NUMBERS B · A
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68 CHAPTER 2 COMPLEX NUMBERS (a) Gi
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70 CHAPTER 2 COMPLEX NUMBERS One so
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72 CHAPTER 2 COMPLEX NUMBERS The
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74 CHAPTER 2 COMPLEX NUMBERS (a) Gi
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76 CHAPTER 2 COMPLEX NUMBERS Figure
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78 CHAPTER 2 COMPLEX NUMBERS Exerci
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80 CHAPTER 2 COMPLEX NUMBERS (a) Us
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82 CHAPTER 2 COMPLEX NUMBERS The Ma
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84 CHAPTER 2 COMPLEX NUMBERS 2.6 Hi
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86 CHAPTER 2 COMPLEX NUMBERS 2.7 St
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88 CHAPTER 2 COMPLEX NUMBERS Sectio
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3 Modular Arithmetic What goes up,
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92 CHAPTER 3 MODULAR ARITHMETIC Not
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94 CHAPTER 3 MODULAR ARITHMETIC On
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96 CHAPTER 3 MODULAR ARITHMETIC □
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98 CHAPTER 3 MODULAR ARITHMETIC Sup
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100 CHAPTER 3 MODULAR ARITHMETIC wa
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102 CHAPTER 3 MODULAR ARITHMETIC 0
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104 CHAPTER 3 MODULAR ARITHMETIC (i
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106 CHAPTER 3 MODULAR ARITHMETIC No
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108 CHAPTER 3 MODULAR ARITHMETIC x
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110 CHAPTER 3 MODULAR ARITHMETIC He
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112 CHAPTER 3 MODULAR ARITHMETIC Be
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114 CHAPTER 3 MODULAR ARITHMETIC (c
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116 CHAPTER 3 MODULAR ARITHMETIC (a
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118 CHAPTER 3 MODULAR ARITHMETIC
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120 CHAPTER 3 MODULAR ARITHMETIC mu
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122 CHAPTER 3 MODULAR ARITHMETIC We
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124 CHAPTER 3 MODULAR ARITHMETIC Mo
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126 CHAPTER 3 MODULAR ARITHMETIC (a
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128 CHAPTER 3 MODULAR ARITHMETIC Fi
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130 CHAPTER 3 MODULAR ARITHMETIC Fi
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132 CHAPTER 3 MODULAR ARITHMETIC 20
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134 CHAPTER 3 MODULAR ARITHMETIC 1:
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136 CHAPTER 3 MODULAR ARITHMETIC }
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138 CHAPTER 3 MODULAR ARITHMETIC Th
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140 CHAPTER 3 MODULAR ARITHMETIC No
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244 CHAPTER 6 FUNCTIONS: BASIC CONC
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264 CHAPTER 7 INTRODUCTION TO CRYPT
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8 Sigma Notation We’re about to s
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300 CHAPTER 8 SIGMA NOTATION And wh
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302 CHAPTER 8 SIGMA NOTATION Applyi
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308 CHAPTER 8 SIGMA NOTATION Exerci
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310 CHAPTER 8 SIGMA NOTATION (c) (d
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348 CHAPTER 8 SIGMA NOTATION This i
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350 CHAPTER 8 SIGMA NOTATION is a r
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354 CHAPTER 8 SIGMA NOTATION differ
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358 CHAPTER 8 SIGMA NOTATION Pluggi
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360 CHAPTER 8 SIGMA NOTATION 8.8 Hi
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362 CHAPTER 8 SIGMA NOTATION 8.9 St
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364 CHAPTER 8 SIGMA NOTATION 2. 3.
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9 Polynomials In this chapter we’
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368 CHAPTER 9 POLYNOMIALS we find:
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370 CHAPTER 9 POLYNOMIALS According
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372 CHAPTER 9 POLYNOMIALS Remark 9.
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376 CHAPTER 9 POLYNOMIALS This is t
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378 CHAPTER 9 POLYNOMIALS 9.4 More
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380 CHAPTER 9 POLYNOMIALS It turns
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382 CHAPTER 9 POLYNOMIALS c 4 = 4
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384 CHAPTER 9 POLYNOMIALS c 4 = 4
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386 CHAPTER 9 POLYNOMIALS and assoc
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388 CHAPTER 9 POLYNOMIALS ( m ) ∑
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390 CHAPTER 9 POLYNOMIALS Now we mu
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392 CHAPTER 9 POLYNOMIALS 9.6 Polyn
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394 CHAPTER 9 POLYNOMIALS (a) x 2 +
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396 CHAPTER 9 POLYNOMIALS It’s te
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398 CHAPTER 9 POLYNOMIALS (b) Use t
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400 CHAPTER 9 POLYNOMIALS If we set
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402 CHAPTER 9 POLYNOMIALS And here
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404 CHAPTER 9 POLYNOMIALS First ( w
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408 CHAPTER 9 POLYNOMIALS Since thi
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10 Symmetries of Plane Figures “I
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414 CHAPTER 10 SYMMETRIES OF PLANE
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11 Permutations ”For the real env
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444 CHAPTER 11 PERMUTATIONS But wha
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446 CHAPTER 11 PERMUTATIONS Is μ =
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448 CHAPTER 11 PERMUTATIONS a bijec
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454 CHAPTER 11 PERMUTATIONS 11.3.2
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456 CHAPTER 11 PERMUTATIONS Exercis
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458 CHAPTER 11 PERMUTATIONS We may
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460 CHAPTER 11 PERMUTATIONS (ii) In
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462 CHAPTER 11 PERMUTATIONS • 4 d
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464 CHAPTER 11 PERMUTATIONS Then it
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466 CHAPTER 11 PERMUTATIONS (a) S 6
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468 CHAPTER 11 PERMUTATIONS (d) Wha
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470 CHAPTER 11 PERMUTATIONS (a) (12
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472 CHAPTER 11 PERMUTATIONS (a) τ
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474 CHAPTER 11 PERMUTATIONS What Pr
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476 CHAPTER 11 PERMUTATIONS (a) (14
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478 CHAPTER 11 PERMUTATIONS 11.5
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480 CHAPTER 11 PERMUTATIONS the per
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482 CHAPTER 11 PERMUTATIONS Figure
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484 CHAPTER 11 PERMUTATIONS 11.6 Ot
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486 CHAPTER 11 PERMUTATIONS Figure
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488 CHAPTER 11 PERMUTATIONS So far
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490 CHAPTER 11 PERMUTATIONS In ligh
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492 CHAPTER 11 PERMUTATIONS 11.7 Ad
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494 CHAPTER 11 PERMUTATIONS 11.8 Hi
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496 CHAPTER 12 INTRODUCTION TO GROU
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13 Further Topics in Cryptography I
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544 CHAPTER 13 FURTHER TOPICS IN CR
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572CHAPTER 14 EQUIVALENCE RELATIONS
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15 Cosets and Quotient Groups (a.k.
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616CHAPTER 15 COSETS AND QUOTIENT G
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652CHAPTER 16 ERROR-DETECTING AND C
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17 Isomorphisms of Groups Thanks to
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696 CHAPTER 17 ISOMORPHISMS OF GROU
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18 Homomorphisms of Groups In this
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746 CHAPTER 18 HOMOMORPHISMS OF GRO
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19 Group Actions We’ve defined a
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766 CHAPTER 19 GROUP ACTIONS Figure
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820 CHAPTER 19 GROUP ACTIONS Proof.
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20 Introduction to Rings and Fields
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832 CHAPTER 20 INTRODUCTION TO RING
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900CHAPTER 21 APPENDIX: INDUCTION P
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922 GFDL LICENSE 1. Applicability A
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924 GFDL LICENSE 3. Copying In Quan
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926 GFDL LICENSE If the Modified Ve
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928 GFDL LICENSE If a section in th
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930 GFDL LICENSE
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932 INDEX Caesar, Julius, 265 Cance
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936 INDEX multiplicative, 38 of a c
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938 INDEX definition, 34 Miller-Rab
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Abstract Algebra: Examples and Appl
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