## Transcription

17 Chapter 17 140, 18 Chapter 18 153, 19 Chapter 19 162. 20 Chapter 20 162, 21 Chapter 21 174, 22 Chapter 22 190. 23 Chapter 24 202, 24 Chapter 25 226, 25 Chapter 32 245. 26 Chapter 33 252, 0 Chapter 0, 0 1 For n 5 8 12 20 and 25 find all positive integers less than n and relatively prime. relatively prime n n, n 12 12 22 3 12 12, 0 2 Determine.

gcd 24 32 5 72 2 33 7 11, lcm 23 32 5 2 33 7 11, 0 3 Determine 51 mod 13. 342 mod 85, 82 73 mod 7, 51 68 mod 7, 35 24 mod 11. and 47 68 mod 11, a mod n a n 17 5 2 17, mod 12 14 14 mod 12 2. 0 4 Find integers s and t such that 1 7 s 11 t Show that s and t are not unique. Find integers s and t such that 1 69 s 31 t, 69 31 2 7 1. 31 7 4 3 2, 18 1 7 3 2, 7 31 7 4 2, 7 31 2 7 8, 69 31 2 9 31 2.

69 9 31 18 31 2, 69 9 31 20, gcd 7 11 gcd 69 31 1, gcd a b 1 s t Z such that as bt 1 4. a b relatively prime gcd a b 1, gcd a b d s t Z such that as bt d 5. 0 6 Suppose a and b are integers that divide the integer c If a and b are relatively. prime show that ab divides c Show by example that if a and b are not relatively. prime then ab need not divide c, Since a c and b c suppose that. c aq1 c bq2 q1 q2 Z 6, Since gcd a b 1, s t Z such that as bt 1. multiplying c, 0 7 If a and b are integers and n is a positive integer prove that a mod n b mod n.

if and only if n divides a b, divide a b a b b a, q b aq a b a 0. a mod n a n a, mod n b mod n a b n a b, n a b mod n Exercise 0 7. a b mod n n a b 7, a mod n b mod n, a n q1 r b n q2 r. a nq1 r b nq2 r q1 q2 Z, nq2 r nq1 nq2 n q1 q2, a b n q1 q2. equivalence relation, 0 8 Let d gcd a b If a da and b db show that gcd a b 1.

Suppose gcd a b k 1 8, a kq1 b kq2 q1 q2 Z, a da dkq1 b. dk gcd a b d, dk d contrary to 8, gcd a b gcd a b, gcd a b gcd a b. Exercise 0 19 9, cyclic group, 0 9 Let n be a fixed positive integer greater than 1 If a mod n a and b mod n b. prove that a b mod n a b mod n and ab mod n a b mod n This. exercise is referred to in Chapter 6 8 10 and 15, a mod n a b mod n b. n a a n b b, a a nq1 b b nq2 q1 q2 Z, a b nq1 a b n q1.

a b a b n q1, a b a b mod n, a b a b mod n, Exercise 0 3. 0 10 Let a and b be positive integers and let d gcd a b and m lcm a b If t divides. both a and b prove that t divides d If s is a multiple of both a and b prove that. s is a multiple of m, Lemma If t a and t b then t as bu for any s u Z. If t a and t b, a tq1 b q1 q2 Z, as t q1 s bu t q2 u q1 q2 s u Z. as bu t q1 s t q2 u t, t as bu 10, s u Z such that as bu d 11. If t a and t b, 0 11 Let n and a be positive integers and let d gcd a n Show that the equation ax.

mod n 1 has a solution if and only if d 1, U n U n Elmentary Number. Theory Z n, Proof We need a lemma, gcd a b 1 s t Z such that as bt 1. It follows immediately from p 4 thm 0 2, If there exists s t Z such that as bt 1 then since gcd a b a and. gcd a b b we have gcd a b as bt 1 Which implies that gcd a b 1. s t Z such that as nt 1, as 1 mod n, ax 1 mod n has a solution. a has a multiplicative inverse modulo n, 0 11 Solve the congruence equation 69x 1 mod 31.

Exercise 0 4, 69 9 31 20 1, 69 9 1 31 20, 69 9 1 mod 31 12. 12 Exercise 0 9 69x 1 mod 31 9, 9 69x 9 1 mod 31, 1 x x 9 mod 31. group inverse, 0 13 Suppose that m and n are relatively prime and r is any integer Show that there. are integers x and y such that mx ny r, Consider the set S ms nt s t Z. Consider the subset S a S a 0 of S, Prove that S, Apply the Well Ordering Principle on S there is a smallest positive integer.

Suppose that d mp nq, If c m and c n by 10 c mp nq d. That is d gcd m n, If gcd m n 1 then there exist p q Z such that mp nq 1 Thus. r m pr n qr Let x pr and y qr, 0 16 Determine 71000 mod 6 and 61001 mod 7. 7 1 mod 6 and 6 1 mod 7 Exercise 0 9, 61001 1 1001 1 6 mod 7. 0 17 Let a b s and t be integers If a mod st b mod st show that a mod s b. mod s and a mod t b mod t What condition on s and t is needed to make. the converse true, 0 18 Determine 8402 mod 5, 82 1 mod 5 9.

8402 82 201 64201 1 201 mod 5, 0 19 Show that gcd a bc 1 if and only if gcd a b 1 and gcd a c 1. Proof gcd a c divides gcd a bc is obviously We show that gcd a bc divides. gcd a c Since gcd a b 1 by p 4 thm 0 2 there exists s t Z such that. Let d gcd a bc, d a and d bc, d a cs bc t as bt c c. 0 19 If gcd a b 1 and a bc then a c, Since a bc suppose that. bc aq q Z 13, Since gcd a b 1, s t Z such that as bt 1. multiplying c, asc aq t c, 0 22 Express 7 3i 1 in standard form.

7 3i 7 3i 7 3i, 0 23 Express 4 5i in standard form. 0 27 For every positive integer n prove that a set with exactly n elements has exactly. 2n subsets counting the empty set and the entire set. 0 30 Generalized Euclid s Lemma If p is a prime and p divides a1 a2 an prove that p. divides ai for some i, 0 31 Use the Generalized Euclid s Lemma see Exercise 0 30 to establish the uniqueness. portion of the Fundamental Theorem of Arithmetic, Use mathematical induction on n. 0 33 Prove that the First Principle of Mathematical Induction isa consequence of the. Well Ordering Principle, 0 37 In the cut As from Songs in the Key of Life Stevie Wonder mentions the equation. 8 8 8 4 Find all integers n for which this statement is true modulo n. n 8 8 8 4 mod n, Stevie Wonder I Just, Called To Say I Love You http goo gl ADsXte.

Stevie Wonder, Who s driving this car Stevie Wonder Stevie Wonder. You Are The Sunshine Of My Life http goo gl BbrnI1. Part Time Lover http goo gl YiLfRe, Sir Duke http goo gl ZxyFeG. Superstition http goo gl W12uEp, Master Blaster Jammin http goo gl tm1xFp. Uptight Everything s Alright http goo gl TxRU5p, 0 39 If it is 2 00 A M now what time will it be 3736 hours from now. 0 50 The 10 digit International Standard Book Number ISBN 10 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10. has the property a1 a2 a10 10 9 8 7 6 5 4 3 2 1 mod 11 0 The digit a10. is the check digit When a10 is required to be 10 to make the dot product 0 the. character X is used as the check digit Suppose that an ISBN 10 has a smudged. entry where the question mark appears in the number 0 716 2841 9 Determine. the missing digit, a1 a2 a10 10 9 8 7 6 5 4 3 2 1, 10 a1 9 a2 8 a3 2 a9 1 a10.

Hardy 1940, 6x 9 mod 11 2, 2 Exercise 0 11, 0 58 Let S be the set of real numbers If a b S define a b if a b is an integer Show. that is an equivalence relation on S Describe the equivalence classes of S. Proof For all a S a a 0 Z so a a or says a a, If a b then a b Z and b a a b Z Thus b a. If a b and b c then a b Z and b c Z and a c a b b c Z. Therefore a c, 3 4 3 4 3 4 4 4 3 4 5 4 3 4, 3 4 a R a 3 4 3 4 3 4 4 4 3 4. 5 4 3 4 3 4 equivalence class, equivalence relation Zn. coset equvalence relation, direct product A B direct product.

A B A B ordered pair, A B a b a A b B, S direct product R2 R R. 13 25 2 2 1 21 10 2, R R subset subset, 13 25 2 2 1. 13 25 2 2 1 R R, a relation on a set S a subset of S S. a b a b 13 25, 13 25 relation, relation equivalence relation. a relation on a set S is called an equivalence relation. if and only if the following statements hold, ii If s t then t s.

iii If s t and t u then s u, ii If s t then t s, iii If s t and t u then s u. equivalence relation relation equivalence rela, tion S a b c d relation on S equivalence relation. a b b c c d a c b d a d, a b b c a c, a b b c a c a a b b c c. a b b c a c a a b b c c d d, a b b c a c a a b b c c d d b a c b c a. the relation on R equivalence relation, the relation on Z equivalence relation equivalence relation.

equivalence relation partition, equivalence relation 13 9. mod 4 13 9 mod 4, Z Z Z Z Z Z, 0 59 Let S be the set of integers If a b S define aRb if ab 0 Is R an equivalence. relation on S, 0 60 Let S be the set of integers If a b S define aRb if a b is even Prove that R is. an equivalence relation and determine the equivalence classes of S. 0 63 What is the last digit of 3100 What is the last digit of 2100. a a mod 10, 0 65 Cancellation Property Suppose and are functions If and is. one to one and onto prove that, 1 Chapter 1, 1 1 With pictures and words describe each symmetry in D3 the set of symmetries of.

an equilateral triangle, 1 2 Write out a complete Cayley table for D3 Is D3 Abelian. abelian Cayley table, dihedral group Dn Cayley Table. Dn 1 a a2 an 1 b ba ba2 ban 1 a n b 2 ab ba 1, 1 a a b ba ba. D3 1 a a2 b ba ba2, 1 1 a a2 b ba ba2, b b ba ba2 1 a a2. D3 1 a a2 b ba ba2, 1 1 a a2 b ba ba2, a a a2 1 ba2 b ba.

a2 a2 1 a ba ba2 b, b b ba ba2 1 a a2, ba ba ba2 b a2 1 a. ba2 ba2 b ba a a2 1, 1 3 In D4 find all elements X such that. 1 4 Describe in pictures or words the elements of D5 symmetries of a regular pentagon. 1 5 For n 3 describe the elements of Dn Hint You will need to consider two. cases n even and n odd How many elements does Dn have. 1 6 In Dn explain geometrically why a reflection followed by a reflection must be a. n rotation reflection, 3 2 2 1 1 2, 1 2 4 1 2 1, 4 3 3 2 3 4. 1 7 In Dn explain geometrically why a rotation followed by a rotation must be a rota. 1 8 In Dn explain geometrically why a rotation and a reflection taken together in either. order must be a reflection, 1 10 If r1 r2 and r3 represent rotations from Dn and f1 f2 and f3 represent reflections. from Dn determine whether r1 r2 f1 r3 f2 f3 r3 is a rotation or a reflection. 1 11 Find elements A B and C in D4 such that AB BC but A C Thus cross. cancellation is not valid, 1 12 Explain what the following diagram proves about the group Dn.

1 13 Describe the symmetries of a nonsquare rectangle Construct the corresponding. Cayley table, 1 14 Describe the symmetries of a parallelogram that is neither a rectangle nor a rhom. bus Describe the symmetries of a rhombus that is not a rectangle. 1 15 Describe the symmetries of a noncircular ellipse Do the same for a hyperbola. 1 17 For each of the snowflakes in the figure find the symmetry group and locate the. axes of reflective symmetry disregard imperfections. https www youtube com watch v fd hb2xzvZI, http goo gl uRFJba 700. http goo gl CjZGqz, goo gl h7pw3, 1 19 Does a fan blade have a cyclic symmetry group or a dihedral symmetry group. 1 20 Bottle caps that are pried off typically have 22 ridges around the rim Find the. symmetry group of such a cap, 2 Chapter 2, Gallian Burton Theorem. exe 0 13 p 21 thm 2 3 gcd a b d s t such that as bt d. p 23 thm 2 4 gcd a b 1 s t such that as bt 1, exe 0 6 p 23 cor 2 gcd a b 1 a c b c ab c.

p 24 thm 2 5 gcd a b 1 a bc a c, p 79 cor 2 a n as e n s. exe 3 4 x x 1, p 80 thm 4 2 ar gcd n r n, 2 1 Which of the following binary operations are closed. a subtraction of positive integers, b division of nonzero integers. c function composition of polynomials with real coefficients. d multiplication of 2 2 matrices with integer entries. 2 2 Which of the following binary operations are associative. a multiplication mod n, b division of nonzero rationals. c function composition of polynomials with real coefficients. d multiplication of 2 2 matrices with integer entries. 2 3 Which of the following binary operations are commutative. a subtraction of integers, b division of nonzero real numbers.

c function composition of polynomials with real coefficients. d multiplication of 2 2 matrices with integer entries. 2 4 Which of the following sets are closed under the given operation. a 0 4 8 12 addition mod 16, b 0 4 8 12 addition mod 15. c 1 4 7 13 multiplication mod 15, d 1 4 5 7 multiplication mod 9. 2 5 In each case find the inverse of the element under the given operation. a 13 in Z20, b 13 in U 14, c n 1 in U n n 2, d 3 2i in C the group of nonzero complex numbers under multiplication. 2 6 In each case perform the indicated operation, a In C 7 5i 3 2i. b In GL 2 Z13 det, c In GL 2 R, d In GL 2 Z13, 2 8 Referring to Example 13 verify the assertion that subtraction is not associative.

2 9 Show that 1 2 3 under multiplication modulo 4 is not a group but that 1 2 3 4. under multiplication modulo 5 is a group, 2 10 Show that the group GL 2 R of Example 9 is non Abelian by exhibiting a pair of. matrices A and B in GL 2 R such that AB BA, 2 12 Given an example of group elements a and b with the property that a 1 ba b. 2 15 Let G be a group and let H x 1 x G Show that G H as sets. Contemporary Abstract Algebra 8 e 0 bfhaha gmail com January 12 2017 Contents 0 Chapter 0 2 1 Chapter 1 12 2 Chapter 2 14 3 Chapter 3 21 4 Chapter 4 36 5 Chapter 5 47 6 Chapter 6 62 7 Chapter 7 66 8 Chapter 8 73 9 Chapter 9 78 10 Chapter 10 93 11 Chapter 11 102 12 Chapter 12 106 13 Chapter 13 110 14 Chapter 14 119 15 Chapter 15 130 16 Chapter 16 135 1 17 Chapter 17 140 18 Chapter 18 153 19