IM Abstract Algebra Manamtam PDF

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Summary

No part of this publication may be reproduced or copied by recording or other electronic/mechanical methods, without the prior written permission of the publisher/compiler [email protected]. Faculty members whose names are printed on the cover are onlycompilers who collected materials from diffe...

Description

No part of this publication may be reproduced or copied by recording or other electronic/ mechanical methods, without the prior written permission of the publisher/compiler via [email protected]. Faculty members whose names are printed on the cover are only compilers who collected materials from different authors. This is not for sale and the compilers have no intention to profit from this.

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INTRODUCTION COURSE OVERVIEW This Instructional Material deals with the study of equivalence relations, mathematical induction and division algorithm. You will be learning the integrals involving groups, subgroups, and cyclic groups.

GRADING SYSTEM The grading system will be as follows. Practice Exercises (Class Standing)

70%

Reflective Journal

30%

FINAL GRADE

100%

REFLECTIVE JOURNAL DIRECTIONS A reflective journal is a place to write down your daily or weekly reflection entries. You can write about a positive or negative event that you experienced, what it means or meant to you, and what you may have learned from that experience. Directions: Following these steps will guide you in our reflective journal requirement. 1. Take a picture of yourself together with what you work on the day you read and work on the exercises. Choose a clear photo or make the photo clear and visible. 2. Create a simple narration on what you feel, what you learn, the difficulties you encounter while reading and answering the exercises, and what you did to overcome those difficulties. 3. Organize and Compile it (in a single file). 4. The number of photos should not be more that 20, assuming you will be working the lessons per week. 5. Prepare a soft copy or hard copy of it depending on your situation, if you have internet connection or none. 6. Submit it together with your answers in modules on the agreed date of submission. 7. Make sure your photo-essay meets the criteria on the rubric. The rubric is provided in the next page.

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TABLE OF CONTENTS TITLE

Pages

Lesson 1:

Preliminaries

1 − 16

Lesson 2:

The Integers

17 − 27

Lesson 3:

Groups

28 − 45

Lesson 4:

Cyclic Groups

46 − 58

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LEARNING MATERIALS IN SEMA 30183: ABSTRACT ALGEBRA Jay-R A. Manamtam Polytechnic University of the Philippines

LESSON 1: PRELIMINARIES LEARNING OBJECTIVES • Review the concept of proofs. • Discuss cartesian products and mappings. • Describe equivalence relations and partitions.

DISCUSSIONS

b c x2 + x = − a a ( )2 ( )2 c b b b − x2 + x + = a a 2a 2a ( )2 2 b b − 4ac x+ = 4a2 2a √ b ± b2 − 4ac x+ = 2a 2a √ −b ± b2 − 4ac x=

∪

Ai = A1 ∪ . . . ∪ An

∩

Ai = A1 ∩ . . . ∩ An

i=1

i=1

A ∪ B = {x ∈ R : 0 < x < 4} A \ B = {x ∈ R : 0 < x < 2}

= {x : x ∈ A} =A

= {x : x ∈ A}

= A ∩ A′ ∩ B ∩ B ′

A

B f b c

A

g

B

b c

g

f

A

X

b

Y

c

Z

g◦f

C

Y Z

√ √ x ) = ( 3 x )3 = x

)( ) ( ax + by x = y c d

π (1) π(2) π (3)

)

=

( 1 2 3

= h(g(f (a))) = (h ◦ g)(f (a))

−5

A

3

)

;

◦ TA (x, y) = TA ◦ T −1

B

(x, y) = (ax + by, cx + dy)

B

(x, y) = (3ax + 3by, 0)

2 3 1

)

3 1 2

)

2 1 + y1

= x22 + y2

x22

+

1

y2

[1] = {. . . , −2, 1, 4, 7, . . .},

+ y12 =

p−2

q2 7q 2

−

p q

1

+ y21 = x22 + y2

Abstract Algebra(2020), by Thomas W. Judson.

LEARNING MATERIALS IN SEMA 30183: ABSTRACT ALGEBRA Jay-R A. Manamtam Polytechnic University of the Philippines

LESSON 2: THE INTEGERS LEARNING OBJECTIVES • Explain the concept of mathematical induction. • Discuss the division algorithm. • Perform Euclidean algorithm. • Apply the fundamental theorem of arithmetic.

DISCUSSIONS

= 10(10k+1 + 3 · 10k + 5) − 45

n ( ∑ n k=0

k

k

k

)

+

(

)

)

=

n! k!(n − k)!

( ) ( n n + = k

) n! n n! = + (k − 1)!(n − k + 1)! k!(n − k)! k−1 (n + 1)! = k!(n + 1 − k)! ( n+1 =

(

) n ( ) ∑ n k n−k = (a + b) a b k k=0

n ∑

n ∑ ( ) ( ) n ak+1bn−k + n ak bn+1−k k k ) k=0 n ( n ( ) k=0 ∑ ∑ n n k n+1−k n+1 k n+1−k =a + a b + a b + bn+1 k−1 k k=1 k=1 ) ( )] n [( ∑ n n ak bn+1−k + bn+1 + = an+1 + k k−1 k=1 n+1 ∑ (n + 1 = k=0

= a − (ar + bs)q

= a − arq − bsq

945 = 525 · 1 + 420 525 = 420 · 1 + 105

= 525 + (−1) · [945 + (−1) · 525]

= 2 · 525 + (−1) · 945 = 2 · [2415 + (−2) · 945] + (−1) · 945

r1 = r2 q 3 + r3

= rn−2 − rn−1 qn

= rn−2 − qn (rn−3 − qn−1 rn−2 ) = −qn rn−3 + (1 + qn qn−1 )rn−2

6

4

2

n

√ 1∑ a1 a2 · · · an ≤ n k=1

n ( ∑ n k=0

2

+

n 1 1 = +··· + n(n + 1) n+1 6

A(x + 1, 0) = A(x, 1),

Abstract Algebra(2020), by Thomas W. Judson.

LEARNING MATERIALS IN SEMA 30183: ABSTRACT ALGEBRA Jay-R A. Manamtam Polytechnic University of the Philippines

LESSON 3: GROUPS LEARNING OBJECTIVES • Discuss integer equivalence classes. • Explain the concept of binary operation. • Discuss basic properties of groups. • Establish the concept of subgroups.

DISCUSSIONS

0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0

0 1 2 3 4 5 6 7

0 2 4 6 0 2 4 6

0 3 6 1 4 7 2 5

0 4 0 4 0 4 0 4

0 5 2 7 4 1 6 3

0 6 4 2 0 6 4 2

0 7 6 5 4 3 2 1

□

D

B

A

B

C

D

C

C

D

C

B

A

B

B

A

C

C

D

B

D

C

C

A

B

B D

D

D

180◦

B

B id =

A B C

C

A B

A

A ρ1 =

(

ρ2 =

(

C B

A

C

C B

A

C

C B

B

C B

A

C

A B C B C A

)

A B C C A B

)

) ( A B C µ1 = A C B

A B C C B A

)

A B C µ3 = B A C

)

µ2 = A

( A B C)

( (

(µ1 ρ1 )(B) = µ1 (ρ1 (B )) = µ1 (C ) = B

0 1 2 3 4

0 1 2 3 4

1 2 3 4 0

2 3 4 0 1

3 4 0 1 2

4 0 1 2 3

□

2·2=4

3·2=0

1 3 5 7

1 3 5 7

3 1 7 5

5 7 1 3

7 5 3 1

□

c d

ad − bc

)

0 1 ( ) 0 i J= i 0

(

)

d

(

−b

0 1 I= −1 0 ( i 0 K=

)

a2 + b2

| {z } |

{z

}

c d

)

0 1

(0, 0) (0, 1) (1, 0) (1, 1)

)

+

( −1 0

(0, 0) (0, 1) (1, 0) (1, 1)

) ( 0 0 0 = −1

(0, 1) (0, 0) (1, 1) (1, 0)

(1, 0) (1, 1) (0, 0) (0, 1)

(1, 1) (1, 0) (0, 1) (0, 0)

□

a b c d

a b c d

c b d a

d a c d a b b c

a b c d

a b c d

b c d a

c d d a a b b c

a a b c d b b a d c c c d a b

a b c d

a b c d

b a b d

c d c d a d b c

 1 x y 0 1 z  0 0 1

    1 x y 1 x′ y ′ 1 x + x′ y + y′ + xz′ 0 1 z  0 1 z ′  =  0 1 z + z′ 0 0 1 0 0 1 0 0 1

n

−1 · · · g −1 gn−1

0

50000

30042

6

Abstract Algebra(2020), by Thomas W. Judson.

LEARNING MATERIALS IN SEMA 30183: ABSTRACT ALGEBRA Jay-R A. Manamtam Polytechnic University of the Philippines

LESSON 4: CYCLIC GROUPS LEARNING OBJECTIVES • Discuss cyclic groups.

• Apply method of repeated squares.

• Explain subgroups of cyclic groups. • Discuss multiplicative group of complex numbers. • Introduced the circle of group and the roots of unity.

DISCUSSIONS

23 = 8

24 = 7

S3

□

7 · 9 = 15 10 · 9 = 10

8·9=8 11 · 9 = 3

−

|z| =

13 √ 13

9·9=1 12 · 9 = 12

3 i 13

z1 = 2 + 3i z3 = −3 + 2i x

0 z2 = 1 − 2i

a + bi r θ 0

x

4

))10 2

2

)

)

√ 2 + 2 i 2√ √ 2 2 ω3 = − i + 2 2 √ √ 2 2 ω5 = − i − √2 √2 2 2 − ω7 = 2

i ω3 x

−1 ω7

ω5 −i

−1 0

)

1

0

)

−1 −1/2

1 −1

)

0

) √ 3/2

)

Abstract Algebra(2020), by Thomas W. Judson....