Discrete and Combinatorial Algebra

John Rickert, Associate Professor of Mathematics

Office hours this week: MTR 7,9, or make an appointment, or drop in.

On some Thursdays I might be proctoring IFYCSEM exams.

The average score on the final excam was 74.7. You may come to my office and pick-up your exam.

A sketch of the answers to Quiz #7 is online.

To Homework ...Questions

Homework for our next class ...Today's questions

Please let me know if I've missed anything.

The main goal in this class is to have you (the student) perform as an active learner. To do this you will need to do the exercises, raise questions about structures that you are studying, create hypotheses and test these hypotheses.

The quizzes, examinations and homework done during the year will be worth 80% of the course grade. The final examination will be worth 20% of the grade.

For Tuesday 12/1: Read Section 4.0 and work the exercises in 4.0.1, 4.0.2.

For Thursday 12/2: Read through section 4.2.2 work the exercises. Think about exercise 4.0.2.1

The average score on Thursday's quiz was 16.7 out of 20

Work the exercises

Work the exercises.

Prove or disprove:

How about :

Read through exercise 4.3.2. Work exercises 4.3.1, 4.3.2. Be sure to be prepared to work exercise 4.3.2

Read through exercise 4.3.2. Work exercises 4.3.1, 4.3.2. Be sure to be prepared to work exercise 4.3.2

Work exercise 4.3.3

Hand in exercise 4.3.2

Work through the work in the reading and do exercises 4.3.4,4.3.5 and 4.3.6

Work exercises 4.3.8.

How many symmetries does a cube have? What is the group of rotational symmetries of a cube?

Hand In Exercise 4.3.8.2

Exam #1, Monday, January 11, 1999

Do exercises 5.0.1, 5.0.2

Read through exercise 5.1.5. Do exercises 5.1.1 - 5.1.5, Especially 5.1.5.

Do the exercises.

Understand how errors are detected and corrected using the code C

Come to class with questions about the chapter 5 material.

Quiz #7 answers

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- their FFPA factorizations correspond to the same partition of 4.
- the first term in the list form of the permutations are equal.
- they have the same order.
- The number of transpositions required to describe the permutations is the same for each element.

Prove your assertion.

Is it possible for ~ to be a congruence relation over one of these (either G,+ or G\0,*) and not the other?

Dennis Lin observes that a~b iff |a|=|b| is an equivalence relation over the integers (and rationals, reals, complex numbers, etc.) so that it is a congruence relation with respect to multiplication, but not addition.

It was observed that the breakdown into "even" and "odd" permutations is a congruence relation over S

How many congruence relations are there over S

Peter Webb noticed that any function f(x) can be used to define an equivalence relation by: a~b <-> f(a)=f(b).

Can you prove that this is an equivalence relation?

Nathan Froyd observed that the sequence begins 1,1,2,5,15,52,203,... and 203=52*1+15*5+5*10+2*10+1*5+1*1.

The number of congruence relations is still an open question.

How many symmetries does a cube have? What is the group of rotational symmetries of a cube?

What is the largest error correcting code on 15 bits?

Here's the (probably inefficient) code that I'm using to get

c.4:=y^2: c.5:=y^2+1: c.6:=y^2+y: c.7:=y^2+y+1:

for k from 8 to 15 do c.k:=y^3+c.(k-8) od:

md:=y^4+y^3+1;

reduce:= poly-> modp(rem(expand(poly),md,y),2);

The matrix, unfortunately, must be typed in. (OK. I admit, there's a sneaky way to do it, but I'd like to leave

If the matrix is called

and determine s

Please inform me if there are any typos that make the code fail to perform.

We saw in class that a

Can we further streamline the calculation of

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A question from class:

Expanding the denominator gives (1-3s)(1-s-s

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