# Abstract Algebra

**Syllabus of Algebra Abstract Course**

download Syllabus abstract algebra IMPoME 2011

download book: Maddox, Randall B – Mathematical Thinking and Writing

**Algebraic and Ordering Properties of R (3# meeting)**

Wednesday, 8th Sep, 2011

In the 3rd meeting, Mr. Darmo start the class by stating that students have to able think mathematically. There two important aspects in assessing students’ ability are the mathematics material itself and the way of student thinking. Mr. Darmo explain that there are 4 properties in the thinking mathematically:

- Logic (have to give reasoning)
- Systematic
- Conceptual
- Analytic

**For example:**

Prove that A ∩ B = A, if A, B, C be number of set and A ⊆ B.

*(the important and first thing that we have to make sure is ‘do the students understand the question/problem’)*

*To solve that problem we have assumption*: A, B, C be number of set and A ⊆ B

*And we have to make conclusion in the end that: * A ∩ B = A

In this case the *keyword* is “=”

Related to the equality (=) we have concept that: A ∩ B = A means A ∩ B ⊆ A and A ⊆ A ∩ B ..**(conceptual)**

**(Analyze the problem):
**

A ∩ B = A means that A ∩ B ⊆ A and A ⊆ A ∩ B

for A ∩ B ⊆ A means ∀x ∈ A ∩ B → x ∈ A and A ⊆ A ∩ B means ∀x ∈ A → x ∈ A ∩ B * (general statement)*

To show that general statement we can use the assumption.

After we can show the proof we have to conclude that A ∩ B = A.

The thinking hierarchy is called **systematic** aspect, while the whole process is** logic.**

**Theorem 2.3.6**

If a, b ∈ R, then

1. (−a)b = −(ab),

2. (−a)(−b) = ab.

We will try to prove 1.

If a, b ∈ R, show that (−a)b = −(ab)

*Idea*: To show the solution we need to show that (-a)b + ab = 0

*We have some concept*: Theorem 2.3.4 For every a ∈ R, −(−a) = a.

*The way of thinking/ structure the solution:*

(-a)b + ab = 0 by assumption A14 then (-a+a)b = 0, then based on theo. 2.3.4 we get 0.b = 0

Solution: because b ∈ R, by using theorem 2.3.2 we have 0.b = 0, (-a+a)b = 0. Accordingly, -a.b + ab = 0. This means that (−a)b = −(ab)……… (proven)

**Question 6.** page 59 *Book; Mathematical Thinking and Writting (wrtten by: Randall B. Maddox)*

Prove the principle of zero product: if ab = 0 then either a=0 or b=0

Keyword: “or’

To show the solution we need to show that for a≠0 then b≠0

proof: let b≠0,

ab = 0

ab . ab(invers) = o. ab

1 = 0 *this contradict with A15 which said that 1*≠0 so this means that b=0

another proof:

ab = 0

ab . invers of a = o. invers of a

a. invers of a .b = 0

1. b = 0

b = 0 (proven)

The advantage to structure the question is:

- it can be use to know students’ thinking
- it can be used as instrument to measure whether the question can be read/understand by students

Mr. Darmo stated that one of the problems in school is students can not solve the given problem or question. This maybe not because students can not solve or do mathematics computation but maybe because the students can not understand/read the question. So that, students should be taught how to read or understand and structure the given problem rather than how to get the final or right answer.

In this course, Mr. Darmo also explain about the relation between dicdactical and mathematics itself. He said that dicdactic is important thing that should be had by teacher. If a teacher doesn’t have dicdactica then he/she will not be able to teach mathematics effectively and understandably. Dedactic lead people tostudy hard and to make learning math become more eassily and interesting. If People do not know about the dicdactic then they will be stress to teach math, but if math teacher have dicdactic but doesn’t understand math, then they will lead their students to the wrong or mistake.