Unit 1: Preliminary Information
Before we can begin looking at algebraic structures, such as groups,
rings, and fields, we must review the information that provides the
basis for those structures. All of these structures have sets, and set
theory applies to groups at their core. Next, we will review the set of
integers, as integers have all the properties that groups possess.
Indeed, the set of integers, *Z*, with the operation “+” is itself a
group.
After discussing sets and operations on them, we will review relations
and functions, as the concept of homomorphisms and isomorphisms arise
from special cases of functions from one group, ring, or field to
another. Finally, we will look at a special class of function known as
the permutation, which plays a major role in group theory.
Unit 1 Time Advisory
This unit will take you 33 hours to complete.
☐ Subunit 1.1: 15 hours
☐ Subunit 1.1.1: 5 hours
☐ Subunit 1.1.2: 5 hours
☐ Subunit 1.1.3: 5 hours
☐ Subunit 1.2: 10 hours
☐ Subunit 1.2.1: 3 hours
☐ Subunit 1.2.2: 3 hours
☐ Subunit 1.2.3: 4 hours
☐ Subunit 1.3: 5 hours
☐ Subunit 1.4: 3 hours
Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define and manipulate sets.
- Use set operations properly.
- Use modulo arithmetic.
- Create and use permutations.
1.1 Basic Set Theory
1.1.1 Definition of Sets
- Reading: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “Sets and Equivalence
Relations”
Link: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “Sets and Equivalence
Relations”
(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)
Instructions: Please click on the link above. For the section on
sets, read “1.2 Sets and Equivalence Relations,” pages 4-5. Please
note that this PDF file will be used for the entire course. It will
be referenced for readings and assignments throughout.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410-417 of the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download.html) (PDF).
Reading: Reading: Wikibooks: “Set Theory/Sets” Link: Wikibooks: “Set Theory/Sets” (PDF)
Instructions: Please read the entire webpage. This link provides considerable, condensed information on sets and set theory. For most, this will be a review of the primary terminology, axioms, and properties of sets.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikibooks version of this article here (HTML).Lecture: Harvard University Extension: Dr. Paul Bamberg’s “Math E-102 Sets, Counting, and Probability”: “Probability, Intuition, and Axioms” Links: Harvard University Extension: Dr. Paul Bamberg’s “Math E-102 Sets, Counting, and Probability”: “Probability, Intuition, and Axioms” (YouTube)
Also Available in: Quicktime, Adobe Flash, or Mp3
Instructions: Please click on the link, and watch the video in its entirety. To choose another format, scroll down the webpage to Lecture 1 and choose the format most appropriate for your internet connection. The video mainly discusses probability, but the 45:30 mark has a useful discussion on set terminology and notation. Dr. Bamberg is a former Rhodes Scholar in theoretical physics and is now a full-time math instructor at Harvard University’s Extension School’s Open Learning Initiative.
Terms of Use: This video has been reposted with permission for non-profit educational use by Harvard University. The original version can be found here.Lecture: Harvard University Extension: Dr. Paul Bamberg’s “Math E-102 Sets, Counting, and Probability”: “Probability by Counting and Inclusion-Exclusion” Link: Harvard University Extension: Dr. Paul Bamberg’s “Math E-102 Sets, Counting, and Probability”: “Probability by Counting and Inclusion-Exclusion” (YouTube)
Also Available in: Quicktime, Adobe Flash, or Mp3
Instructions: Please click on the link, and watch the video in its entirety. To choose another format, scroll down the webpage to Lecture 2 and choose the format most appropriate for your internet connection. The video clip discusses DeMorgan’s laws and gives an example of a Venn diagram. Dr. Bamberg is a former Rhodes Scholar in theoretical physics and is now a full-time math instructor at Harvard University’s Extension School’s Open Learning Initiative.
Terms of Use: This video has been reposted with permission for non-profit educational use by Harvard University. The original version can be found here.Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 1 and 2” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 1 and 2” (PDF)
Also available in:Instructions: Work through problems 1 and 2 on page 18. After you have completed each problem, check your answers with the solutions on page 395.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410-417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
1.1.2 Examples of Sets
- Reading: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “Sets and Equivalence
Relations”
Link: Stephen F. Austin State University: Thomas W.
Judson’s Abstract Algebra Theory and Applications: “Sets and
Equivalence
Relations”
(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)
Instructions: For the section on examples of sets, read “1.2 Sets
and Equivalence Relations,” page 5. This PDF file will be used for
the entire course. It will be referenced for readings and
assignments throughout.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410-417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata-20100827.pdf) (PDF).
- Lecture: YouTube: Xoax.net’s “Algebra 1: Simple Sets”
Link: YouTube: Xoax.net's “Algebra 1: Simple
Sets” (YouTube)
Instructions: Please click on the link, and view this brief video in its entirety (3:58 minutes) to learn about types of sets and ways to describe sets.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
1.1.3 Set Operations
- Reading: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “Sets and Equivalence
Relations”
Link: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and
Applications: “Sets
and Equivalence Relations” (PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)
Instructions: For the section on examples of sets, read “1.2 Sets
and Equivalence Relations,” pages 5-8. Please note that page 5 is a
review of the information you read in subunit 1.2.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410-417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata-20100827.pdf) (PDF).
Lecture: YouTube: APUS07’s “Intersection, Union, and Complement” Link: YouTube: APUS07’s “Intersection, Union, and Complement” (YouTube)
Instructions: Please click on the link, and view the brief video lecture in its entirety (2:50 minutes) for a discussion on sets and common set operations.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 6, 10, and 14” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 6, 10, and 14” (PDF)
Also available in:Instructions: Please work through problems 6, 10, and 14 on page 19. After you have completed all of the assigned problems, check your answers against the solutions on page 395.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410-417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
1.2 Integers
1.2.1 Divisors
- Reading: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “The Integers”
Link: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “The
Integers”
(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)
Instructions: For the section on examples of sets, read “2.2 The
Division Algorithm,” on pages 26-29.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410-417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata-20100827.pdf) (PDF).
Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Integers”: “Exercise Problems 19 and 23” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Integers”: “Exercise Problems 19 and 23” (PDF)
Also available in:Instructions: Complete problems 19 and 23 on page 33. Then, check your answers against the solutions on page 396.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410-417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
1.2.2 Primes
- Reading: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “The Integers”
Link: Stephen F. Austin State University: Thomas W.
Judson’s Abstract Algebra Theory and Applications: “The
Integers”
(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)
Instructions: For the section on examples of sets, read “2.2 The
Division Algorithm,” pages 29-30.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410-417. The material linked above is
licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata-20100827.pdf) (PDF).
Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Integers”: “Exercise Problems 27 and 29” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Integers”: “Exercise Problems 27 and 29” (PDF)
Also available in:Instructions: Complete problems 27 and 29 on page 33. Then, check your answers against the solutions on page 396.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410-417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
1.2.3 Congruences and Integers Modulo
- Reading: Northern Illinois University: John A. Beachy’s and
William D. Blair’s Abstract Algebra: “Integers”: “Congruences”
Link: Northern Illinois University: John A. Beachy’s and William D.
Blair’s Abstract Algebra: “Integers”:
“Congruences”
(HTML)
Instructions: Please click the link above, and read the text in its
entirety. The page defines and discusses modulo n. Dr. John A.
Beachy is Professor Emeritus of Mathematical Sciences at Northern
Illinois University.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
Lecture: Harvard University Extension: Dr. Benedict Gross’s “Math E-222 Abstract Algebra”: “Congruence mod n” Link: Harvard University Extension: Dr. Benedict Gross’s “Math E-222 Abstract Algebra”: “Congruence mod n” (YouTube)
Also Available in: Adobe Flash, Quicktime, or Mp3
Instructions: Please click on the link, and watch the video. To choose a different format, scroll down to Lecture 6 (Week 2, Lecture 3) and choose the format most appropriate for your internet connection. The video clip ties integers mod n to cosets, which helps with the understanding of cosets later. Please watch the entire video (about 52 minutes).
Terms of Use: This video has been reposted with permission for non-profit educational use by Harvard University. The original version can be found here.Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problem 1” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problem 1” (PDF)
Also available in:Instructions: Complete problem 1 on page 49. Then, check your answer with the solution on page 397.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410-417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
1.3 Relations and Functions
- Reading: Stephen F. Austin State University: Thomas W. Judson’s
Abstract Algebra Theory and Applications: “Sets and Equivalence
Relations”
Link: Stephen F. Austin State University: Thomas W. Judson’s
“Abstract Algebra Theory and
Applications”
“Sets and Equivalence Relations” (PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)
Instructions: For the section on examples of sets, read “1.2 Sets
and Equivalence Relations,” pages 8-15.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410-417 of the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata-20100827.pdf) (PDF).
Lecture: YouTube: ClevelandMath’s “Relations and Functions” Link: YouTube: ClevelandMath’s “Relations and Functions” (YouTube)
Instructions: Please click on the link, and view the video in its entirety (6:51 minutes). In this video lecture, Mr. Durbin discusses relations and functions, using various common notations. Since abstract algebra is largely about properties of sets with operators acting on them, this is a good introduction.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 18, 20, and 22” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Preliminaries”: “Exercise Problems 18, 20, and 22” (PDF)
Also available in:Instructions: Work through problems 18, 20, and 22 on page 19. Then, check your answers with the solutions on page 395.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410-417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
1.4 Permutations
- Reading: Wolfram MathWorld: Eric W. Weisstein’s “Permutation”
Link: Wolfram MathWorld: Eric. W. Weisstein’s
“Permutation”
(HTML)
Instructions: Please click on the link to read the information on
the entire webpage. This webpage contains a concise rendering of
information on general permutations. For those who have had several
classes in math, this represents a good refresher on the topic.
Terms of Use: Please respect Wolfram MathWorld's terms of
use. MathWorld
webpages are free for academic use and may be hyperlinked, according
to their FAQ
site.
- Lecture: YouTube: Burny1’s “Group Theory Permutations”
Link: YouTube: Burny1’s “Group Theory
Permutations”
(YouTube)
Instructions: Please click on the link, and view the brief video in its entirety (3:41 minutes). The video discusses general permutations and then discusses permutations in light of groups. This is a good introduction to the concept.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.